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Metamath Proof Explorer

Theorem sb2ae

Description: In the case of two successive substitutions for two always equal variables, the second substitution has no effect. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ and WL, 9-Aug-2023) (New usage is discouraged.)

		Ref	Expression
	Assertion	sb2ae	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑣 / 𝑦 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		drsb1	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑢 / 𝑦 ] [ 𝑣 / 𝑦 ] 𝜑 ) )
2		nfs1v	⊢ Ⅎ 𝑦 [ 𝑣 / 𝑦 ] 𝜑
3	2	sbf	⊢ ( [ 𝑢 / 𝑦 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑣 / 𝑦 ] 𝜑 )
4	1 3	bitrdi	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑣 / 𝑦 ] 𝜑 ) )