Theorem frege57b

Description: Analogue of frege57aid . Proposition 57 of Frege1879 p. 51. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege57b	⊢ ( 𝑥 = 𝑦 → ( [ 𝑦 / 𝑧 ] 𝜑 → [ 𝑥 / 𝑧 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		frege52b	⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑧 ] 𝜑 → [ 𝑥 / 𝑧 ] 𝜑 ) )
2		frege56b	⊢ ( ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑧 ] 𝜑 → [ 𝑥 / 𝑧 ] 𝜑 ) ) → ( 𝑥 = 𝑦 → ( [ 𝑦 / 𝑧 ] 𝜑 → [ 𝑥 / 𝑧 ] 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( 𝑥 = 𝑦 → ( [ 𝑦 / 𝑧 ] 𝜑 → [ 𝑥 / 𝑧 ] 𝜑 ) )