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Metamath Proof Explorer

Theorem sbequi

Description: An equality theorem for substitution. (Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 15-Sep-2018) (Proof shortened by Steven Nguyen, 7-Jul-2023)

		Ref	Expression
	Assertion	sbequi	⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sbequ	⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 ↔ [ 𝑦 / 𝑧 ] 𝜑 ) )
2	1	biimpd	⊢ ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) )