Metamath Proof Explorer

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	$⊢ [y/ x] φ \to (y = z \to [z/ x] φ)$

Proof

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