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Theorem pm14.122b 38941
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 2662 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 329 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 1889 . . . 4 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
4 dfsbcq 3470 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
54bibi1d 332 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
63, 5imbi12d 333 . . 3 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑)) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑))))
7 sbc5 3493 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
8 nfa1 2068 . . . . 5 𝑥𝑥(𝜑𝑥 = 𝑦)
9 simpr 476 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝜑)
10 ancr 571 . . . . . . 7 ((𝜑𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦𝜑)))
1110sps 2093 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦𝜑)))
129, 11impbid2 216 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ((𝑥 = 𝑦𝜑) ↔ 𝜑))
138, 12exbid 2129 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥𝜑))
147, 13syl5bb 272 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑))
156, 14vtoclg 3297 . 2 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
1615pm5.32d 672 1 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-sbc 3469
This theorem is referenced by:  pm14.122c  38942
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