Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  or2expropbilem2 Structured version   Visualization version   GIF version

Theorem or2expropbilem2 47045
Description: Lemma 2 for or2expropbi 47046 and ich2exprop 47458. (Contributed by AV, 16-Jul-2023.)
Assertion
Ref Expression
or2expropbilem2 (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 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   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)

Proof of Theorem or2expropbilem2
StepHypRef Expression
1 nfv 1914 . 2 𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
2 nfv 1914 . 2 𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
3 nfv 1914 . . 3 𝑎𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
4 nfcv 2905 . . . 4 𝑎𝑦
5 nfsbc1v 3808 . . . 4 𝑎[𝑥 / 𝑎]𝜑
64, 5nfsbcw 3810 . . 3 𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
73, 6nfan 1899 . 2 𝑎(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
8 nfv 1914 . . 3 𝑏𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
9 nfsbc1v 3808 . . 3 𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
108, 9nfan 1899 . 2 𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
11 opeq12 4875 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
1211eqeq2d 2748 . . 3 ((𝑎 = 𝑥𝑏 = 𝑦) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩))
13 sbceq1a 3799 . . . 4 (𝑎 = 𝑥 → (𝜑[𝑥 / 𝑎]𝜑))
14 sbceq1a 3799 . . . 4 (𝑏 = 𝑦 → ([𝑥 / 𝑎]𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
1513, 14sylan9bb 509 . . 3 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
1612, 15anbi12d 632 . 2 ((𝑎 = 𝑥𝑏 = 𝑦) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
171, 2, 7, 10, 16cbvex2v 2346 1 (∃𝑎𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  [wsbc 3788  cop 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  or2expropbi  47046  ich2exprop  47458
  Copyright terms: Public domain W3C validator