Theorem frege53c

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

	Ref	Expression
Assertion	frege53c	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → ( 𝐴 = 𝐵 → [ 𝐵 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege52c	⊢ ( 𝐴 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐵 / 𝑥 ] 𝜑 ) )
2		ax-frege8	⊢ ( ( 𝐴 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜑 → [ 𝐵 / 𝑥 ] 𝜑 ) ) → ( [ 𝐴 / 𝑥 ] 𝜑 → ( 𝐴 = 𝐵 → [ 𝐵 / 𝑥 ] 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → ( 𝐴 = 𝐵 → [ 𝐵 / 𝑥 ] 𝜑 ) )