Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ich2exprop Structured version   Visualization version   GIF version

Theorem ich2exprop 44156
 Description: If the setvar variables are interchangeable in a wff, there is an ordered pair fulfilling the wff iff there is an unordered pair fulfilling the wff. (Contributed by AV, 16-Jul-2023.)
Assertion
Ref Expression
ich2exprop ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏   𝑋,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)

Proof of Theorem ich2exprop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . . 5 𝑎 𝐴𝑋
2 nfv 1915 . . . . 5 𝑎 𝐵𝑋
3 nfich1 44132 . . . . 5 𝑎[𝑎𝑏]𝜑
41, 2, 3nf3an 1902 . . . 4 𝑎(𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑)
5 nfv 1915 . . . . . . 7 𝑎𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
6 nfcv 2955 . . . . . . . 8 𝑎𝑦
7 nfsbc1v 3742 . . . . . . . 8 𝑎[𝑥 / 𝑎]𝜑
86, 7nfsbcw 3744 . . . . . . 7 𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
95, 8nfan 1900 . . . . . 6 𝑎(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
109nfex 2332 . . . . 5 𝑎𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
1110nfex 2332 . . . 4 𝑎𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
12 nfv 1915 . . . . . 6 𝑏 𝐴𝑋
13 nfv 1915 . . . . . 6 𝑏 𝐵𝑋
14 nfich2 44133 . . . . . 6 𝑏[𝑎𝑏]𝜑
1512, 13, 14nf3an 1902 . . . . 5 𝑏(𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑)
16 nfv 1915 . . . . . . . 8 𝑏𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
17 nfsbc1v 3742 . . . . . . . 8 𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
1816, 17nfan 1900 . . . . . . 7 𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
1918nfex 2332 . . . . . 6 𝑏𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
2019nfex 2332 . . . . 5 𝑏𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
21 vex 3445 . . . . . . . . 9 𝑎 ∈ V
22 vex 3445 . . . . . . . . 9 𝑏 ∈ V
23 preq12bg 4747 . . . . . . . . 9 (((𝐴𝑋𝐵𝑋) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎))))
2421, 22, 23mpanr12 704 . . . . . . . 8 ((𝐴𝑋𝐵𝑋) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎))))
25243adant3 1129 . . . . . . 7 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ↔ ((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎))))
26 or2expropbilem1 43792 . . . . . . . . 9 ((𝐴𝑋𝐵𝑋) → ((𝐴 = 𝑎𝐵 = 𝑏) → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
27263adant3 1129 . . . . . . . 8 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → ((𝐴 = 𝑎𝐵 = 𝑏) → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
28 ichcom 44144 . . . . . . . . . . . . . . 15 ([𝑎𝑏]𝜑 ↔ [𝑏𝑎]𝜑)
2928biimpi 219 . . . . . . . . . . . . . 14 ([𝑎𝑏]𝜑 → [𝑏𝑎]𝜑)
30293ad2ant3 1132 . . . . . . . . . . . . 13 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → [𝑏𝑎]𝜑)
3130adantr 484 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) ∧ 𝜑) → [𝑏𝑎]𝜑)
3222, 21pm3.2i 474 . . . . . . . . . . . . 13 (𝑏 ∈ V ∧ 𝑎 ∈ V)
3332a1i 11 . . . . . . . . . . . 12 ((𝐴 = 𝑏𝐵 = 𝑎) → (𝑏 ∈ V ∧ 𝑎 ∈ V))
3431, 33anim12i 615 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) ∧ 𝜑) ∧ (𝐴 = 𝑏𝐵 = 𝑎)) → ([𝑏𝑎]𝜑 ∧ (𝑏 ∈ V ∧ 𝑎 ∈ V)))
35 simpr 488 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) ∧ 𝜑) → 𝜑)
36 opeq12 4771 . . . . . . . . . . . 12 ((𝐴 = 𝑏𝐵 = 𝑎) → ⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩)
3735, 36anim12ci 616 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) ∧ 𝜑) ∧ (𝐴 = 𝑏𝐵 = 𝑎)) → (⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑))
38 nfv 1915 . . . . . . . . . . . 12 𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑)
39 nfv 1915 . . . . . . . . . . . 12 𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑)
40 opeq12 4771 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑏𝑦 = 𝑎) → ⟨𝑥, 𝑦⟩ = ⟨𝑏, 𝑎⟩)
4140eqeq2d 2809 . . . . . . . . . . . . . 14 ((𝑥 = 𝑏𝑦 = 𝑎) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩))
4241adantl 485 . . . . . . . . . . . . 13 (([𝑏𝑎]𝜑 ∧ (𝑥 = 𝑏𝑦 = 𝑎)) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩))
43 dfsbcq 3724 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑[𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
4443adantl 485 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑏𝑦 = 𝑎) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑[𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
4544adantl 485 . . . . . . . . . . . . . 14 (([𝑏𝑎]𝜑 ∧ (𝑥 = 𝑏𝑦 = 𝑎)) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑[𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
46 sbceq1a 3733 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑏 → ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
4746adantr 484 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑏𝑦 = 𝑎) → ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑))
48 df-ich 44131 . . . . . . . . . . . . . . . 16 ([𝑏𝑎]𝜑 ↔ ∀𝑏𝑎([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑𝜑))
49 sbsbc 3726 . . . . . . . . . . . . . . . . . 18 ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
50 sbsbc 3726 . . . . . . . . . . . . . . . . . . . 20 ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑[𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
51 sbsbc 3726 . . . . . . . . . . . . . . . . . . . . 21 ([𝑥 / 𝑎]𝜑[𝑥 / 𝑎]𝜑)
5251sbcbii 3778 . . . . . . . . . . . . . . . . . . . 20 ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑[𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
5350, 52bitri 278 . . . . . . . . . . . . . . . . . . 19 ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑[𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
5453sbcbii 3778 . . . . . . . . . . . . . . . . . 18 ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
5549, 54bitri 278 . . . . . . . . . . . . . . . . 17 ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑)
56 2sp 2183 . . . . . . . . . . . . . . . . 17 (∀𝑏𝑎([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑𝜑) → ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑𝜑))
5755, 56bitr3id 288 . . . . . . . . . . . . . . . 16 (∀𝑏𝑎([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑𝜑) → ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑𝜑))
5848, 57sylbi 220 . . . . . . . . . . . . . . 15 ([𝑏𝑎]𝜑 → ([𝑏 / 𝑥][𝑎 / 𝑏][𝑥 / 𝑎]𝜑𝜑))
5947, 58sylan9bbr 514 . . . . . . . . . . . . . 14 (([𝑏𝑎]𝜑 ∧ (𝑥 = 𝑏𝑦 = 𝑎)) → ([𝑎 / 𝑏][𝑥 / 𝑎]𝜑𝜑))
6045, 59bitrd 282 . . . . . . . . . . . . 13 (([𝑏𝑎]𝜑 ∧ (𝑥 = 𝑏𝑦 = 𝑎)) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑𝜑))
6142, 60anbi12d 633 . . . . . . . . . . . 12 (([𝑏𝑎]𝜑 ∧ (𝑥 = 𝑏𝑦 = 𝑎)) → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑)))
6238, 39, 61spc2ed 3551 . . . . . . . . . . 11 (([𝑏𝑎]𝜑 ∧ (𝑏 ∈ V ∧ 𝑎 ∈ V)) → ((⟨𝐴, 𝐵⟩ = ⟨𝑏, 𝑎⟩ ∧ 𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
6334, 37, 62sylc 65 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) ∧ 𝜑) ∧ (𝐴 = 𝑏𝐵 = 𝑎)) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
6463exp31 423 . . . . . . . . 9 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (𝜑 → ((𝐴 = 𝑏𝐵 = 𝑎) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
6564com23 86 . . . . . . . 8 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → ((𝐴 = 𝑏𝐵 = 𝑎) → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
6627, 65jaod 856 . . . . . . 7 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (((𝐴 = 𝑎𝐵 = 𝑏) ∨ (𝐴 = 𝑏𝐵 = 𝑎)) → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
6725, 66sylbid 243 . . . . . 6 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
6867impd 414 . . . . 5 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
6915, 20, 68exlimd 2216 . . . 4 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (∃𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
704, 11, 69exlimd 2216 . . 3 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
71 or2expropbilem2 43793 . . 3 (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
7270, 71syl6ibr 255 . 2 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) → ∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
73 oppr 43790 . . . . 5 ((𝐴𝑋𝐵𝑋) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → {𝐴, 𝐵} = {𝑎, 𝑏}))
7473anim1d 613 . . . 4 ((𝐴𝑋𝐵𝑋) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)))
75742eximdv 1920 . . 3 ((𝐴𝑋𝐵𝑋) → (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)))
76753adant3 1129 . 2 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑)))
7772, 76impbid 215 1 ((𝐴𝑋𝐵𝑋 ∧ [𝑎𝑏]𝜑) → (∃𝑎𝑏({𝐴, 𝐵} = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781  [wsb 2069   ∈ wcel 2111  Vcvv 3442  [wsbc 3722  {cpr 4530  ⟨cop 4534  [wich 44130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-ich 44131 This theorem is referenced by: (None)
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