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Theorem 2sbc6g 39221
Description: Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
2sbc6g ((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 2sbc6g
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2776 . . . . . . 7 (𝑦 = 𝐵 → (𝑤 = 𝑦𝑤 = 𝐵))
21anbi2d 622 . . . . . 6 (𝑦 = 𝐵 → ((𝑧 = 𝑥𝑤 = 𝑦) ↔ (𝑧 = 𝑥𝑤 = 𝐵)))
32imbi1d 332 . . . . 5 (𝑦 = 𝐵 → (((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑)))
432albidv 2018 . . . 4 (𝑦 = 𝐵 → (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑)))
5 dfsbcq 3598 . . . . 5 (𝑦 = 𝐵 → ([𝑦 / 𝑤]𝜑[𝐵 / 𝑤]𝜑))
65sbcbidv 3651 . . . 4 (𝑦 = 𝐵 → ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑[𝑥 / 𝑧][𝐵 / 𝑤]𝜑))
74, 6bibi12d 336 . . 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 2776 . . . . . . 7 (𝑥 = 𝐴 → (𝑧 = 𝑥𝑧 = 𝐴))
98anbi1d 623 . . . . . 6 (𝑥 = 𝐴 → ((𝑧 = 𝑥𝑤 = 𝐵) ↔ (𝑧 = 𝐴𝑤 = 𝐵)))
109imbi1d 332 . . . . 5 (𝑥 = 𝐴 → (((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑) ↔ ((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)))
11102albidv 2018 . . . 4 (𝑥 = 𝐴 → (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)))
12 dfsbcq 3598 . . . 4 (𝑥 = 𝐴 → ([𝑥 / 𝑧][𝐵 / 𝑤]𝜑[𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
1311, 12bibi12d 336 . . 3 (𝑥 = 𝐴 → ((∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑) ↔ [𝑥 / 𝑧][𝐵 / 𝑤]𝜑) ↔ (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑)))
14 vex 3353 . . . . 5 𝑥 ∈ V
1514sbc6 3623 . . . 4 ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ ∀𝑧(𝑧 = 𝑥[𝑦 / 𝑤]𝜑))
16 19.21v 2034 . . . . . 6 (∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦𝜑)) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦𝜑)))
17 impexp 441 . . . . . . 7 (((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (𝑧 = 𝑥 → (𝑤 = 𝑦𝜑)))
1817albii 1914 . . . . . 6 (∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦𝜑)))
19 vex 3353 . . . . . . . 8 𝑦 ∈ V
2019sbc6 3623 . . . . . . 7 ([𝑦 / 𝑤]𝜑 ↔ ∀𝑤(𝑤 = 𝑦𝜑))
2120imbi2i 327 . . . . . 6 ((𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦𝜑)))
2216, 18, 213bitr4ri 295 . . . . 5 ((𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ ∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
2322albii 1914 . . . 4 (∀𝑧(𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
2415, 23bitr2i 267 . . 3 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑)
257, 13, 24vtocl2g 3422 . 2 ((𝐵𝐷𝐴𝐶) → (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
2625ancoms 450 1 ((𝐴𝐶𝐵𝐷) → (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  [wsbc 3596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-sbc 3597
This theorem is referenced by:  pm14.123a  39231
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