This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 )