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Theorem or2expropbilem2 46983
Description: Lemma 2 for or2expropbi 46984 and ich2exprop 47396. (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 1912 . 2 𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
2 nfv 1912 . 2 𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)
3 nfv 1912 . . 3 𝑎𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
4 nfcv 2903 . . . 4 𝑎𝑦
5 nfsbc1v 3811 . . . 4 𝑎[𝑥 / 𝑎]𝜑
64, 5nfsbcw 3813 . . 3 𝑎[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
73, 6nfan 1897 . 2 𝑎(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
8 nfv 1912 . . 3 𝑏𝐴, 𝐵⟩ = ⟨𝑥, 𝑦
9 nfsbc1v 3811 . . 3 𝑏[𝑦 / 𝑏][𝑥 / 𝑎]𝜑
108, 9nfan 1897 . 2 𝑏(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑)
11 opeq12 4880 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑦) → ⟨𝑎, 𝑏⟩ = ⟨𝑥, 𝑦⟩)
1211eqeq2d 2746 . . 3 ((𝑎 = 𝑥𝑏 = 𝑦) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩))
13 sbceq1a 3802 . . . 4 (𝑎 = 𝑥 → (𝜑[𝑥 / 𝑎]𝜑))
14 sbceq1a 3802 . . . 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 2345 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 1537  wex 1776  [wsbc 3791  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  or2expropbi  46984  ich2exprop  47396
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