Theorem frege53c

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

	Ref	Expression
Assertion	frege53c	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(A=B\to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-frege52c	$$⊢A=B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$
2		ax-frege8	$$⊢\left(A=B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(A=B\to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)\right)$$
3	1 2	ax-mp	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(A=B\to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$