Theorem frege57c 37234

Description: Swap order of implication in ax-frege52c 37202. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege57c.a	⊢ 𝐴 ∈ 𝐶

Assertion

Ref	Expression
frege57c	⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem frege57c

Step	Hyp	Ref	Expression
1		ax-frege52c 37202	. 2 ⊢ (𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
2		frege57c.a	. . 3 ⊢ 𝐴 ∈ 𝐶
3	2	frege56c 37233	. 2 ⊢ ((𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) → (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)))
4	1, 3	ax-mp 5	1 ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege52c 37202

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-sn 4126

This theorem is referenced by: (None)