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Theorem or2expropbilem2 43551
Description: Lemma 2 for or2expropbi 43552 and ich2exprop 43914. (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 1916 . 2 𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
2 nfv 1916 . 2 𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
3 nfv 1916 . . 3 𝑎𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
4 nfcv 2982 . . . 4 𝑎𝑦
5 nfsbc1v 3778 . . . 4 𝑎[𝑥 / 𝑎]𝜑
64, 5nfsbcw 3780 . . 3 𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
73, 6nfan 1901 . 2 𝑎(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
8 nfv 1916 . . 3 𝑏𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
9 nfsbc1v 3778 . . 3 𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
108, 9nfan 1901 . 2 𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
11 opeq12 4791 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
1211eqeq2d 2835 . . 3 ((𝑎 = 𝑥𝑏 = 𝑦) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩))
13 sbceq1a 3769 . . . 4 (𝑎 = 𝑥 → (𝜑[𝑥 / 𝑎]𝜑))
14 sbceq1a 3769 . . . 4 (𝑏 = 𝑦 → ([𝑥 / 𝑎]𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
1513, 14sylan9bb 513 . . 3 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜑))
1612, 15anbi12d 633 . 2 ((𝑎 = 𝑥𝑏 = 𝑦) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)))
171, 2, 7, 10, 16cbvex2v 2367 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 209  wa 399   = wceq 1538  wex 1781  [wsbc 3758  cop 4556
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-sbc 3759  df-un 3924  df-sn 4551  df-pr 4553  df-op 4557
This theorem is referenced by:  or2expropbi  43552  ich2exprop  43914
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