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Theorem 2sbc6g 41520
 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 2770 . . . . . . 7 (𝑦 = 𝐵 → (𝑤 = 𝑦𝑤 = 𝐵))
21anbi2d 631 . . . . . 6 (𝑦 = 𝐵 → ((𝑧 = 𝑥𝑤 = 𝑦) ↔ (𝑧 = 𝑥𝑤 = 𝐵)))
32imbi1d 345 . . . . 5 (𝑦 = 𝐵 → (((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑)))
432albidv 1924 . . . 4 (𝑦 = 𝐵 → (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑)))
5 dfsbcq 3700 . . . . 5 (𝑦 = 𝐵 → ([𝑦 / 𝑤]𝜑[𝐵 / 𝑤]𝜑))
65sbcbidv 3753 . . . 4 (𝑦 = 𝐵 → ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑[𝑥 / 𝑧][𝐵 / 𝑤]𝜑))
74, 6bibi12d 349 . . 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 2770 . . . . . . 7 (𝑥 = 𝐴 → (𝑧 = 𝑥𝑧 = 𝐴))
98anbi1d 632 . . . . . 6 (𝑥 = 𝐴 → ((𝑧 = 𝑥𝑤 = 𝐵) ↔ (𝑧 = 𝐴𝑤 = 𝐵)))
109imbi1d 345 . . . . 5 (𝑥 = 𝐴 → (((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑) ↔ ((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)))
11102albidv 1924 . . . 4 (𝑥 = 𝐴 → (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝐵) → 𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑)))
12 dfsbcq 3700 . . . 4 (𝑥 = 𝐴 → ([𝑥 / 𝑧][𝐵 / 𝑤]𝜑[𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
1311, 12bibi12d 349 . . 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 3413 . . . . 5 𝑥 ∈ V
1514sbc6 3729 . . . 4 ([𝑥 / 𝑧][𝑦 / 𝑤]𝜑 ↔ ∀𝑧(𝑧 = 𝑥[𝑦 / 𝑤]𝜑))
16 19.21v 1940 . . . . . 6 (∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦𝜑)) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦𝜑)))
17 impexp 454 . . . . . . 7 (((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (𝑧 = 𝑥 → (𝑤 = 𝑦𝜑)))
1817albii 1821 . . . . . 6 (∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦𝜑)))
19 vex 3413 . . . . . . . 8 𝑦 ∈ V
2019sbc6 3729 . . . . . . 7 ([𝑦 / 𝑤]𝜑 ↔ ∀𝑤(𝑤 = 𝑦𝜑))
2120imbi2i 339 . . . . . 6 ((𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦𝜑)))
2216, 18, 213bitr4ri 307 . . . . 5 ((𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ ∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
2322albii 1821 . . . 4 (∀𝑧(𝑧 = 𝑥[𝑦 / 𝑤]𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
2415, 23bitr2i 279 . . 3 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ [𝑥 / 𝑧][𝑦 / 𝑤]𝜑)
257, 13, 24vtocl2g 3492 . 2 ((𝐵𝐷𝐴𝐶) → (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
2625ancoms 462 1 ((𝐴𝐶𝐵𝐷) → (∀𝑧𝑤((𝑧 = 𝐴𝑤 = 𝐵) → 𝜑) ↔ [𝐴 / 𝑧][𝐵 / 𝑤]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  [wsbc 3698 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-sbc 3699 This theorem is referenced by:  pm14.123a  41530
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