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 → [ 𝐴 / 𝑧 ] 𝑧 = 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2	1	frege54cor1c	⊢ [ 𝑥 / 𝑧 ] 𝑧 = 𝑥
3		frege53c	⊢ ( [ 𝑥 / 𝑧 ] 𝑧 = 𝑥 → ( 𝑥 = 𝐴 → [ 𝐴 / 𝑧 ] 𝑧 = 𝑥 ) )
4	2 3	ax-mp	⊢ ( 𝑥 = 𝐴 → [ 𝐴 / 𝑧 ] 𝑧 = 𝑥 )