Theorem frege56c

Description: Lemma for frege57c . Proposition 56 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege56c.b	$$⊢B\in C$$
Assertion	frege56c	$$⊢\left(A=B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)\right)\to \left(B=A\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege56c.b	$$⊢B\in C$$
2	1	frege54cor1c	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}x=B$$
3		frege53c	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}x=B\to \left(B=A\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)$$
4	2 3	ax-mp	$$⊢B=A\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B$$
5		frege55lem1c	$$⊢\left(B=A\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}x=B\right)\to \left(B=A\to A=B\right)$$
6	4 5	ax-mp	$$⊢B=A\to A=B$$
7		frege9	$$⊢\left(B=A\to A=B\right)\to \left(\left(A=B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)\right)\to \left(B=A\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)\right)\right)$$
8	6 7	ax-mp	$$⊢\left(A=B\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)\right)\to \left(B=A\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)\right)$$