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	$$⊢A\in C$$
Assertion	frege57c	$$⊢A=B\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		frege57c.a	$$⊢A\in C$$
2		ax-frege52c	$$⊢B=A\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
3	1	frege56c	$$⊢\left(B=A\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)\to \left(A=B\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)$$
4	2 3	ax-mp	$$⊢A=B\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$