Theorem pm13.13a 39573

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	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑)

Proof of Theorem pm13.13a

Step	Hyp	Ref	Expression
1		sbceq1a 3663	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	biimpac 472	1 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601  [wsbc 3652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-sbc 3653

This theorem is referenced by:  pm13.194  39578