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Metamath Proof Explorer

Theorem sb3

Description: One direction of a simplified definition of substitution when variables are distinct. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 21-Feb-2024) (New usage is discouraged.)

		Ref	Expression
	Assertion	sb3	$⊢ \neg \forall x x = y \to (\exists x (x = y \land φ) \to [y/ x] φ)$

Proof

Step	Hyp	Ref	Expression
1		sb3b	$⊢ \neg \forall x x = y \to ([y/ x] φ \leftrightarrow \exists x (x = y \land φ))$
2	1	biimprd	$⊢ \neg \forall x x = y \to (\exists x (x = y \land φ) \to [y/ x] φ)$