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Theorem pm14.122a 40909
 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 1890 . 2 (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝑥 = 𝐴𝜑)))
2 sbc6g 3781 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
32bicomd 225 . . 3 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ [𝐴 / 𝑥]𝜑))
43anbi2d 630 . 2 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝑥 = 𝐴𝜑)) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
51, 4syl5bb 285 1 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  [wsbc 3752 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3475  df-sbc 3753 This theorem is referenced by:  pm14.122c  40911
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