Theorem frege53b

Description: Lemma for frege102 (via frege92 ). Proposition 53 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege53b	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( 𝑦 = 𝑧 → [ 𝑧 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		frege52b	⊢ ( 𝑦 = 𝑧 → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑧 / 𝑥 ] 𝜑 ) )
2		ax-frege8	⊢ ( ( 𝑦 = 𝑧 → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑧 / 𝑥 ] 𝜑 ) ) → ( [ 𝑦 / 𝑥 ] 𝜑 → ( 𝑦 = 𝑧 → [ 𝑧 / 𝑥 ] 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ( 𝑦 = 𝑧 → [ 𝑧 / 𝑥 ] 𝜑 ) )