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Metamath Proof Explorer

Theorem clelsb1

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 ). (Contributed by Rodolfo Medina, 28-Apr-2010) (Proof shortened by Andrew Salmon, 14-Jun-2011)

		Ref	Expression
	Assertion	clelsb1	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eleq1w	⊢ ( 𝑥 = 𝑤 → ( 𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴 ) )
2		eleq1w	⊢ ( 𝑤 = 𝑦 → ( 𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
3	1 2	sbievw2	⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 )