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Theorem pm14.122a 42684
Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.122a (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem pm14.122a
StepHypRef Expression
1 albiim 1892 . 2 (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝑥 = 𝐴𝜑)))
2 sbc6g 3769 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
32bicomd 222 . . 3 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ [𝐴 / 𝑥]𝜑))
43anbi2d 629 . 2 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝑥 = 𝐴𝜑)) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
51, 4bitrid 282 1 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  [wsbc 3739
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-sbc 3740
This theorem is referenced by:  pm14.122c  42686
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