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Metamath Proof Explorer

Theorem equsalh

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsalhw for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 2-Jun-1993) (New usage is discouraged.)

	Ref	Expression
Hypotheses	equsalh.1	$⊢ ψ \to \forall x ψ$
	equsalh.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
Assertion	equsalh	$⊢ \forall x (x = y \to φ) \leftrightarrow ψ$

Proof

Step	Hyp	Ref	Expression
1		equsalh.1	$⊢ ψ \to \forall x ψ$
2		equsalh.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
3	1	nf5i	$⊢ Ⅎ x ψ$
4	3 2	equsal	$⊢ \forall x (x = y \to φ) \leftrightarrow ψ$