Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  or2expropbilem1 Structured version   Visualization version   GIF version

Theorem or2expropbilem1 43990
 Description: Lemma 1 for or2expropbi 43992 and ich2exprop 44356. (Contributed by AV, 16-Jul-2023.)
Assertion
Ref Expression
or2expropbilem1 ((𝐴𝑋𝐵𝑋) → ((𝐴 = 𝑎𝐵 = 𝑏) → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑎,𝑏,𝑥,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝑋(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem or2expropbilem1
StepHypRef Expression
1 vex 3413 . . . . . . . 8 𝑎 ∈ V
2 vex 3413 . . . . . . . 8 𝑏 ∈ V
31, 2pm3.2i 474 . . . . . . 7 (𝑎 ∈ V ∧ 𝑏 ∈ V)
43a1i 11 . . . . . 6 ((𝐴𝑋𝐵𝑋) → (𝑎 ∈ V ∧ 𝑏 ∈ V))
54anim1ci 618 . . . . 5 (((𝐴𝑋𝐵𝑋) ∧ 𝜑) → (𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)))
65adantr 484 . . . 4 ((((𝐴𝑋𝐵𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎𝐵 = 𝑏)) → (𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)))
7 sbcid 3713 . . . . . . 7 ([𝑏 / 𝑏][𝑎 / 𝑎]𝜑[𝑎 / 𝑎]𝜑)
8 sbcid 3713 . . . . . . 7 ([𝑎 / 𝑎]𝜑𝜑)
97, 8sylbbr 239 . . . . . 6 (𝜑[𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
109adantl 485 . . . . 5 (((𝐴𝑋𝐵𝑋) ∧ 𝜑) → [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
11 opeq12 4765 . . . . 5 ((𝐴 = 𝑎𝐵 = 𝑏) → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩)
1210, 11anim12ci 616 . . . 4 ((((𝐴𝑋𝐵𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎𝐵 = 𝑏)) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
13 nfv 1915 . . . . 5 𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
14 nfv 1915 . . . . 5 𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
15 opeq12 4765 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑏) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)
1615eqeq2d 2769 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
17 dfsbcq 3698 . . . . . . . 8 (𝑦 = 𝑏 → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑏][𝑥 / 𝑎]𝜑))
18 dfsbcq 3698 . . . . . . . . 9 (𝑥 = 𝑎 → ([𝑥 / 𝑎]𝜑[𝑎 / 𝑎]𝜑))
1918sbcbidv 3751 . . . . . . . 8 (𝑥 = 𝑎 → ([𝑏 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
2017, 19sylan9bbr 514 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
2116, 20anbi12d 633 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)))
2221adantl 485 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)))
2313, 14, 22spc2ed 3520 . . . 4 ((𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
246, 12, 23sylc 65 . . 3 ((((𝐴𝑋𝐵𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎𝐵 = 𝑏)) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
2524exp31 423 . 2 ((𝐴𝑋𝐵𝑋) → (𝜑 → ((𝐴 = 𝑎𝐵 = 𝑏) → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
2625com23 86 1 ((𝐴𝑋𝐵𝑋) → ((𝐴 = 𝑎𝐵 = 𝑏) → (𝜑 → ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3409  [wsbc 3696  ⟨cop 4528 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-12 2175  ax-ext 2729 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 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-sbc 3697  df-un 3863  df-sn 4523  df-pr 4525  df-op 4529 This theorem is referenced by:  or2expropbi  43992  ich2exprop  44356
 Copyright terms: Public domain W3C validator