Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  br4 Structured version   Visualization version   GIF version

Theorem br4 32891
Description: Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
Hypotheses
Ref Expression
br4.1 (𝑎 = 𝐴 → (𝜑𝜓))
br4.2 (𝑏 = 𝐵 → (𝜓𝜒))
br4.3 (𝑐 = 𝐶 → (𝜒𝜃))
br4.4 (𝑑 = 𝐷 → (𝜃𝜏))
br4.5 (𝑥 = 𝑋𝑃 = 𝑄)
br4.6 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}
Assertion
Ref Expression
br4 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜏))
Distinct variable groups:   𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥,𝐴   𝐵,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥   𝜒,𝑏   𝑄,𝑎,𝑏,𝑐,𝑑,𝑥   𝐶,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥   𝐷,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥   𝜓,𝑎   𝑋,𝑎,𝑏,𝑐,𝑑,𝑥   𝑃,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞   𝑆,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑥   𝜏,𝑎,𝑏,𝑐,𝑑,𝑥   𝜃,𝑐   𝜑,𝑝,𝑞,𝑥
Allowed substitution hints:   𝜑(𝑎,𝑏,𝑐,𝑑)   𝜓(𝑥,𝑞,𝑝,𝑏,𝑐,𝑑)   𝜒(𝑥,𝑞,𝑝,𝑎,𝑐,𝑑)   𝜃(𝑥,𝑞,𝑝,𝑎,𝑏,𝑑)   𝜏(𝑞,𝑝)   𝑃(𝑥)   𝑄(𝑞,𝑝)   𝑅(𝑥,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝑋(𝑞,𝑝)

Proof of Theorem br4
StepHypRef Expression
1 opex 5347 . . 3 𝐴, 𝐵⟩ ∈ V
2 opex 5347 . . 3 𝐶, 𝐷⟩ ∈ V
3 eqeq1 2822 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
433anbi1d 1431 . . . . . 6 (𝑝 = ⟨𝐴, 𝐵⟩ → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
54rexbidv 3294 . . . . 5 (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
652rexbidv 3297 . . . 4 (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
762rexbidv 3297 . . 3 (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
8 eqeq1 2822 . . . . . . 7 (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩))
983anbi2d 1432 . . . . . 6 (𝑞 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
109rexbidv 3294 . . . . 5 (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
11102rexbidv 3297 . . . 4 (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
12112rexbidv 3297 . . 3 (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
13 br4.6 . . 3 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}
141, 2, 7, 12, 13brab 5421 . 2 (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
15 vex 3495 . . . . . . . . . . . 12 𝑎 ∈ V
16 vex 3495 . . . . . . . . . . . 12 𝑏 ∈ V
1715, 16opth 5359 . . . . . . . . . . 11 (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐵))
18 br4.1 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (𝜑𝜓))
19 br4.2 . . . . . . . . . . . 12 (𝑏 = 𝐵 → (𝜓𝜒))
2018, 19sylan9bb 510 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝜑𝜒))
2117, 20sylbi 218 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → (𝜑𝜒))
2221eqcoms 2826 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))
23 vex 3495 . . . . . . . . . . . 12 𝑐 ∈ V
24 vex 3495 . . . . . . . . . . . 12 𝑑 ∈ V
2523, 24opth 5359 . . . . . . . . . . 11 (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑐 = 𝐶𝑑 = 𝐷))
26 br4.3 . . . . . . . . . . . 12 (𝑐 = 𝐶 → (𝜒𝜃))
27 br4.4 . . . . . . . . . . . 12 (𝑑 = 𝐷 → (𝜃𝜏))
2826, 27sylan9bb 510 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝜒𝜏))
2925, 28sylbi 218 . . . . . . . . . 10 (⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ → (𝜒𝜏))
3029eqcoms 2826 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ → (𝜒𝜏))
3122, 30sylan9bb 510 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩) → (𝜑𝜏))
3231biimp3a 1460 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏)
3332a1i 11 . . . . . 6 (((((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ (𝑥𝑆𝑎𝑃)) ∧ (𝑏𝑃𝑐𝑃)) ∧ 𝑑𝑃) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏))
3433rexlimdva 3281 . . . . 5 ((((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ (𝑥𝑆𝑎𝑃)) ∧ (𝑏𝑃𝑐𝑃)) → (∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏))
3534rexlimdvva 3291 . . . 4 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ (𝑥𝑆𝑎𝑃)) → (∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏))
3635rexlimdvva 3291 . . 3 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏))
37 simpl1 1183 . . . . 5 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝑋𝑆)
38 simpl2l 1218 . . . . . 6 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝐴𝑄)
39 simpl2r 1219 . . . . . 6 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝐵𝑄)
40 simpl3l 1220 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝐶𝑄)
41 simpl3r 1221 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝐷𝑄)
42 eqidd 2819 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩)
43 eqidd 2819 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
44 simpr 485 . . . . . . 7 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → 𝜏)
45 opeq1 4795 . . . . . . . . . 10 (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
4645eqeq2d 2829 . . . . . . . . 9 (𝑐 = 𝐶 → (⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩))
4746, 263anbi23d 1430 . . . . . . . 8 (𝑐 = 𝐶 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ∧ 𝜃)))
48 opeq2 4796 . . . . . . . . . 10 (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
4948eqeq2d 2829 . . . . . . . . 9 (𝑑 = 𝐷 → (⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩))
5049, 273anbi23d 1430 . . . . . . . 8 (𝑑 = 𝐷 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ∧ 𝜃) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝜏)))
5147, 50rspc2ev 3632 . . . . . . 7 ((𝐶𝑄𝐷𝑄 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝜏)) → ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒))
5240, 41, 42, 43, 44, 51syl113anc 1374 . . . . . 6 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒))
53 opeq1 4795 . . . . . . . . . 10 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5453eqeq2d 2829 . . . . . . . . 9 (𝑎 = 𝐴 → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩))
5554, 183anbi13d 1429 . . . . . . . 8 (𝑎 = 𝐴 → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓)))
56552rexbidv 3297 . . . . . . 7 (𝑎 = 𝐴 → (∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓)))
57 opeq2 4796 . . . . . . . . . 10 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
5857eqeq2d 2829 . . . . . . . . 9 (𝑏 = 𝐵 → (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
5958, 193anbi13d 1429 . . . . . . . 8 (𝑏 = 𝐵 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)))
60592rexbidv 3297 . . . . . . 7 (𝑏 = 𝐵 → (∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓) ↔ ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)))
6156, 60rspc2ev 3632 . . . . . 6 ((𝐴𝑄𝐵𝑄 ∧ ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)) → ∃𝑎𝑄𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
6238, 39, 52, 61syl3anc 1363 . . . . 5 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ∃𝑎𝑄𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
63 br4.5 . . . . . . 7 (𝑥 = 𝑋𝑃 = 𝑄)
6463rexeqdv 3414 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
6563, 64rexeqbidv 3400 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
6663, 65rexeqbidv 3400 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
6763, 66rexeqbidv 3400 . . . . . 6 (𝑥 = 𝑋 → (∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑎𝑄𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
6867rspcev 3620 . . . . 5 ((𝑋𝑆 ∧ ∃𝑎𝑄𝑏𝑄𝑐𝑄𝑑𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)) → ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
6937, 62, 68syl2anc 584 . . . 4 (((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) ∧ 𝜏) → ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))
7069ex 413 . . 3 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (𝜏 → ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)))
7136, 70impbid 213 . 2 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ 𝜏))
7214, 71syl5bb 284 1 ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  cop 4563   class class class wbr 5057  {copab 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator