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 → ( 𝜑𝜓 ) )
Assertion equsalh ( ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ↔ 𝜓 )

Proof

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