Theorem frege53c 37125

Description: Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege53c	⊢ ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵 → [𝐵 / 𝑥]𝜑))

Proof of Theorem frege53c

Step	Hyp	Ref	Expression
1		ax-frege52c 37099	. 2 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
2		ax-frege8 37020	. 2 ⊢ ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵 → [𝐵 / 𝑥]𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵 → [𝐵 / 𝑥]𝜑))

Syntax hints:   → wi 4   = wceq 1474  [wsbc 3306

This theorem was proved from axioms:  ax-mp 5  ax-frege8 37020  ax-frege52c 37099

This theorem is referenced by:  frege55lem2c  37128  frege55c  37129  frege56c  37130  frege92  37166