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Theorem pm14.122c 44448
Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.122c (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem pm14.122c
StepHypRef Expression
1 pm14.122a 44446 . 2 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
2 pm14.122b 44447 . 2 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
31, 2bitrd 279 1 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wex 1778  wcel 2107  [wsbc 3787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-sbc 3788
This theorem is referenced by: (None)
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