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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 → ∀ 𝑥 𝜓 )
	equsalh.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsalh	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsalh.1	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
2		equsalh.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	1	nf5i	⊢ Ⅎ 𝑥 𝜓
4	3 2	equsal	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )