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Metamath Proof Explorer

Theorem 2sbievw

Description: Conversion of double implicit substitution to explicit substitution. Version of 2sbiev with more disjoint variable conditions, requiring fewer axioms. (Contributed by AV, 29-Jul-2023) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

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

Proof

Step	Hyp	Ref	Expression
1		2sbievw.1	⊢ ( ( 𝑥 = 𝑡 ∧ 𝑦 = 𝑢 ) → ( 𝜑 ↔ 𝜓 ) )
2	1	sbiedvw	⊢ ( 𝑥 = 𝑡 → ( [ 𝑢 / 𝑦 ] 𝜑 ↔ 𝜓 ) )
3	2	sbievw	⊢ ( [ 𝑡 / 𝑥 ] [ 𝑢 / 𝑦 ] 𝜑 ↔ 𝜓 )