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Theorem pm13.13a 43468
Description: One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.13a ((𝜑𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑)

Proof of Theorem pm13.13a
StepHypRef Expression
1 sbceq1a 3787 . 2 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
21biimpac 477 1 ((𝜑𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  [wsbc 3776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-sbc 3777
This theorem is referenced by:  pm13.194  43473
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