Theorem frege57c

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

	Ref	Expression
Hypothesis	frege57c.a	⊢ 𝐴 ∈ 𝐶
Assertion	frege57c	⊢ ( 𝐴 = 𝐵 → ( [ 𝐵 / 𝑥 ] 𝜑 → [ 𝐴 / 𝑥 ] 𝜑 ) )

Proof

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