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Metamath Proof Explorer

Theorem ceqsalg

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT . (Contributed by NM, 29-Oct-2003) (Proof shortened by BJ, 29-Sep-2019)

	Ref	Expression
Hypotheses	ceqsalg.1	⊢ Ⅎ 𝑥 𝜓
	ceqsalg.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsalg	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ceqsalg.1	⊢ Ⅎ 𝑥 𝜓
2		ceqsalg.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3	2	ax-gen	⊢ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4		ceqsalt	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝑉 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
5	1 3 4	mp3an12	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )