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Theorem pm14.122b 41124
Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.122b (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem pm14.122b
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2813 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 344 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 1921 . . . 4 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
4 dfsbcq 3725 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
54bibi1d 347 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
63, 5imbi12d 348 . . 3 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑)) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑))))
7 sbc5 3751 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
8 nfa1 2153 . . . . 5 𝑥𝑥(𝜑𝑥 = 𝑦)
9 simpr 488 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝜑)
10 ancr 550 . . . . . . 7 ((𝜑𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦𝜑)))
1110sps 2183 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦𝜑)))
129, 11impbid2 229 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ((𝑥 = 𝑦𝜑) ↔ 𝜑))
138, 12exbid 2224 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥𝜑))
147, 13syl5bb 286 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑))
156, 14vtoclg 3518 . 2 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
1615pm5.32d 580 1 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2112  [wsbc 3723
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 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-sbc 3724
This theorem is referenced by:  pm14.122c  41125
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