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Theorem 2sbc5g 40615
Description: Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
2sbc5g ((𝐴𝐶𝐵𝐷) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
Distinct variable groups:   𝑧,𝑤,𝐴   𝑤,𝐵,𝑧
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐶(𝑧,𝑤)   𝐷(𝑧,𝑤)

Proof of Theorem 2sbc5g
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2837 . . . . . . 7 (𝑦 = 𝐵 → (𝑤 = 𝑦𝑤 = 𝐵))
21anbi2d 628 . . . . . 6 (𝑦 = 𝐵 → ((𝑧 = 𝑥𝑤 = 𝑦) ↔ (𝑧 = 𝑥𝑤 = 𝐵)))
32anbi1d 629 . . . . 5 (𝑦 = 𝐵 → (((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝐵) ∧ 𝜑)))
432exbidv 1918 . . . 4 (𝑦 = 𝐵 → (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) ∧ 𝜑)))
5 dfsbcq 3777 . . . . 5 (𝑦 = 𝐵 → ([𝑦 / 𝑤]𝜑[𝐵 / 𝑤]𝜑))
65sbcbidv 3830 . . . 4 (𝑦 = 𝐵 → ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑[𝑥 / 𝑧][𝐵 / 𝑤]𝜑))
74, 6bibi12d 347 . . 3 (𝑦 = 𝐵 → ((∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑) ↔ (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) ∧ 𝜑) ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑)))
8 eqeq2 2837 . . . . . . 7 (𝑥 = 𝐴 → (𝑧 = 𝑥𝑧 = 𝐴))
98anbi1d 629 . . . . . 6 (𝑥 = 𝐴 → ((𝑧 = 𝑥𝑤 = 𝐵) ↔ (𝑧 = 𝐴𝑤 = 𝐵)))
109anbi1d 629 . . . . 5 (𝑥 = 𝐴 → (((𝑧 = 𝑥𝑤 = 𝐵) ∧ 𝜑) ↔ ((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑)))
11102exbidv 1918 . . . 4 (𝑥 = 𝐴 → (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) ∧ 𝜑) ↔ ∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑)))
12 dfsbcq 3777 . . . 4 (𝑥 = 𝐴 → ([𝑥 / 𝑧][𝐵 / 𝑤]𝜑[𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
1311, 12bibi12d 347 . . 3 (𝑥 = 𝐴 → ((∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) ∧ 𝜑) ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
14 sbc5 3803 . . . 4 ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ ∃𝑧(𝑧 = 𝑥[𝑦 / 𝑤]𝜑))
15 19.42v 1947 . . . . . 6 (∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦𝜑)))
16 anass 469 . . . . . . 7 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦𝜑)))
1716exbii 1841 . . . . . 6 (∃𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦𝜑)))
18 sbc5 3803 . . . . . . 7 ([𝑦 / 𝑤]𝜑 ↔ ∃𝑤(𝑤 = 𝑦𝜑))
1918anbi2i 622 . . . . . 6 ((𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦𝜑)))
2015, 17, 193bitr4ri 305 . . . . 5 ((𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ ∃𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
2120exbii 1841 . . . 4 (∃𝑧(𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
2214, 21bitr2i 277 . . 3 (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑)
237, 13, 22vtocl2g 3576 . 2 ((𝐵𝐷𝐴𝐶) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
2423ancoms 459 1 ((𝐴𝐶𝐵𝐷) → (∃𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) ∧ 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  [wsbc 3775
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-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-v 3501  df-sbc 3776
This theorem is referenced by:  pm14.123b  40625
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