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Metamath Proof Explorer

Theorem ceqsalALT

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. Shorter proof uses df-clab . (Contributed by NM, 18-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ceqsal.1 ⊢ Ⅎ x ψ
ceqsal.2 ⊢ A ∈ V
ceqsal.3 ⊢ x = A → φ ↔ ψ
Assertion ceqsalALT ⊢ ∀ x x = A → φ ↔ ψ

Proof

Step Hyp Ref Expression
1 ceqsal.1 ⊢ Ⅎ x ψ
2 ceqsal.2 ⊢ A ∈ V
3 ceqsal.3 ⊢ x = A → φ ↔ ψ
4 1 3 ceqsalg ⊢ A ∈ V → ∀ x x = A → φ ↔ ψ
5 2 4 ax-mp ⊢ ∀ x x = A → φ ↔ ψ