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Theorem pm14.122b 37442
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 2620 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 328 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 1835 . . . 4 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
4 dfsbcq 3403 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
54bibi1d 331 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
63, 5imbi12d 332 . . 3 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑)) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑))))
7 sbc5 3426 . . . 4 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
8 nfa1 2014 . . . . 5 𝑥𝑥(𝜑𝑥 = 𝑦)
9 simpr 475 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝜑)
10 ancr 569 . . . . . . 7 ((𝜑𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦𝜑)))
1110sps 2042 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦𝜑)))
129, 11impbid2 214 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → ((𝑥 = 𝑦𝜑) ↔ 𝜑))
138, 12exbid 2077 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥𝜑))
147, 13syl5bb 270 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑))
156, 14vtoclg 3238 . 2 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
1615pm5.32d 668 1 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-v 3174  df-sbc 3402
This theorem is referenced by:  pm14.122c  37443
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