Metamath Proof Explorer


Theorem ceqsal

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993)

Ref Expression
Hypotheses ceqsal.1 𝑥 𝜓
ceqsal.2 𝐴 ∈ V
ceqsal.3 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
Assertion ceqsal ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) ↔ 𝜓 )

Proof

Step Hyp Ref Expression
1 ceqsal.1 𝑥 𝜓
2 ceqsal.2 𝐴 ∈ V
3 ceqsal.3 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
4 1 3 ceqsalg ( 𝐴 ∈ V → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) ↔ 𝜓 ) )
5 2 4 ax-mp ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) ↔ 𝜓 )