Theorem frege57c 43244

Description: Swap order of implication in ax-frege52c 43212. 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 43212	. 2 ⊢ (𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
2		frege57c.a	. . 3 ⊢ 𝐴 ∈ 𝐶
3	2	frege56c 43243	. 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 1533   ∈ wcel 2098  [wsbc 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-frege1 43114  ax-frege2 43115  ax-frege8 43133  ax-frege52c 43212

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-sbc 3773  df-sn 4624

This theorem is referenced by: (None)