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Theorem or2expropbilem1 43129
Description: Lemma 1 for or2expropbi 43131 and ich2exprop 43461. (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 3502 . . . . . . . 8 𝑎 ∈ V
2 vex 3502 . . . . . . . 8 𝑏 ∈ V
31, 2pm3.2i 471 . . . . . . 7 (𝑎 ∈ V ∧ 𝑏 ∈ V)
43a1i 11 . . . . . 6 ((𝐴𝑋𝐵𝑋) → (𝑎 ∈ V ∧ 𝑏 ∈ V))
54anim1ci 615 . . . . 5 (((𝐴𝑋𝐵𝑋) ∧ 𝜑) → (𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)))
65adantr 481 . . . 4 ((((𝐴𝑋𝐵𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎𝐵 = 𝑏)) → (𝜑 ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)))
7 sbcid 3792 . . . . . . 7 ([𝑏 / 𝑏][𝑎 / 𝑎]𝜑[𝑎 / 𝑎]𝜑)
8 sbcid 3792 . . . . . . 7 ([𝑎 / 𝑎]𝜑𝜑)
97, 8sylbbr 237 . . . . . 6 (𝜑[𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
109adantl 482 . . . . 5 (((𝐴𝑋𝐵𝑋) ∧ 𝜑) → [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
11 opeq12 4803 . . . . 5 ((𝐴 = 𝑎𝐵 = 𝑏) → ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩)
1210, 11anim12ci 613 . . . 4 ((((𝐴𝑋𝐵𝑋) ∧ 𝜑) ∧ (𝐴 = 𝑎𝐵 = 𝑏)) → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
13 nfv 1908 . . . . 5 𝑥(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
14 nfv 1908 . . . . 5 𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)
15 opeq12 4803 . . . . . . . 8 ((𝑥 = 𝑎𝑦 = 𝑏) → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)
1615eqeq2d 2836 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩))
17 dfsbcq 3777 . . . . . . . 8 (𝑦 = 𝑏 → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑏][𝑥 / 𝑎]𝜑))
18 dfsbcq 3777 . . . . . . . . 9 (𝑥 = 𝑎 → ([𝑥 / 𝑎]𝜑[𝑎 / 𝑎]𝜑))
1918sbcbidv 3830 . . . . . . . 8 (𝑥 = 𝑎 → ([𝑏 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
2017, 19sylan9bbr 511 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ([𝑦 / 𝑏][𝑥 / 𝑎]𝜑[𝑏 / 𝑏][𝑎 / 𝑎]𝜑))
2116, 20anbi12d 630 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑦 / 𝑏][𝑥 / 𝑎]𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ [𝑏 / 𝑏][𝑎 / 𝑎]𝜑)))
2221adantl 482 . . . . 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 3605 . . . 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 420 . 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 207  wa 396   = wceq 1530  wex 1773  wcel 2107  Vcvv 3499  [wsbc 3775  cop 4569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570
This theorem is referenced by:  or2expropbi  43131  ich2exprop  43461
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