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Metamath Proof Explorer

Theorem sbelx

Description: Elimination of substitution. Also see sbel2x . (Contributed by NM, 5-Aug-1993) Avoid ax-13 . (Revised by Wolf Lammen, 6-Aug-2023) Avoid ax-10 . (Revised by GG, 20-Aug-2023)

		Ref	Expression
	Assertion	sbelx	⊢ ( 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ [ 𝑥 / 𝑦 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbequ12r	⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑦 ] 𝜑 ↔ 𝜑 ) )
2	1	equsexvw	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ [ 𝑥 / 𝑦 ] 𝜑 ) ↔ 𝜑 )
3	2	bicomi	⊢ ( 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ [ 𝑥 / 𝑦 ] 𝜑 ) )