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Theorem pm14.122b 39150
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 2782 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 329 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 2001 . . . 4 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
4 dfsbcq 3589 . . . . 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 3612 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
8 nfa1 2184 . . . . 5 𝑥𝑥(𝜑𝑥 = 𝑦)
9 simpr 471 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝜑)
10 ancr 536 . . . . . . 7 ((𝜑𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦𝜑)))
1110sps 2209 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦𝜑)))
129, 11impbid2 216 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ((𝑥 = 𝑦𝜑) ↔ 𝜑))
138, 12exbid 2247 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥𝜑))
147, 13syl5bb 272 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑))
156, 14vtoclg 3417 . 2 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
1615pm5.32d 566 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 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by:  pm14.122c  39151
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