Theorem frege55lem2c

Description: Core proof of Proposition 55 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege55lem2c	$$⊢x=A\to \underset{\dot{}}{[}A/z\underset{\dot{}}{]}z=x$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2	1	frege54cor1c	$$⊢\underset{\dot{}}{[}x/z\underset{\dot{}}{]}z=x$$
3		frege53c	$$⊢\underset{\dot{}}{[}x/z\underset{\dot{}}{]}z=x\to \left(x=A\to \underset{\dot{}}{[}A/z\underset{\dot{}}{]}z=x\right)$$
4	2 3	ax-mp	$$⊢x=A\to \underset{\dot{}}{[}A/z\underset{\dot{}}{]}z=x$$