Theorem pm13.13b 45653

Description: Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.13b	⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑)

Proof of Theorem pm13.13b

Step	Hyp	Ref	Expression
1		sbceq1a 3753	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	biimparc 485	1 ⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sbc 3743

This theorem is used by:  pm14.24  45676