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Metamath Proof Explorer

Theorem eqvinc

Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Proof shortened by Thierry Arnoux, 23-Jan-2022)

		Ref	Expression
	Hypothesis	eqvinc.1	⊢ 𝐴 ∈ V
	Assertion	eqvinc	⊢ ( 𝐴 = 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqvinc.1	⊢ 𝐴 ∈ V
2		eqvincg	⊢ ( 𝐴 ∈ V → ( 𝐴 = 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 = 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) )