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Metamath Proof Explorer

Theorem 2sbiev

Description: Conversion of double implicit substitution to explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See 2sbievw for a version with extra disjoint variables, but based on fewer axioms. (Contributed by AV, 29-Jul-2023) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	2sbiev.1	⊢ ( ( 𝑥 = 𝑡 ∧ 𝑦 = 𝑢 ) → ( 𝜑 ↔ 𝜓 ) )
	Assertion	2sbiev	⊢ ( [ 𝑡 / 𝑥 ] [ 𝑢 / 𝑦 ] 𝜑 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		2sbiev.1	⊢ ( ( 𝑥 = 𝑡 ∧ 𝑦 = 𝑢 ) → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑥 𝜓
3	1	sbiedv	⊢ ( 𝑥 = 𝑡 → ( [ 𝑢 / 𝑦 ] 𝜑 ↔ 𝜓 ) )
4	2 3	sbie	⊢ ( [ 𝑡 / 𝑥 ] [ 𝑢 / 𝑦 ] 𝜑 ↔ 𝜓 )