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Metamath Proof Explorer

Theorem sbtrt

Description: Partially closed form of sbtr . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ, 4-Jun-2019) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sbtrt.nf	⊢ Ⅎ 𝑦 𝜑
	Assertion	sbtrt	⊢ ( ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbtrt.nf	⊢ Ⅎ 𝑦 𝜑
2		stdpc4	⊢ ( ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑥 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 )
3	1	sbid2	⊢ ( [ 𝑥 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 )
4	2 3	sylib	⊢ ( ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → 𝜑 )