Theorem sbtrt 2515

Description: Partially closed form of sbtr 2516. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by BJ, 4-Jun-2019.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sbtrt.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sbtrt	⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑)

Proof of Theorem sbtrt

Step	Hyp	Ref	Expression
1		stdpc4 2071	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
2		sbtrt.nf	. . 3 ⊢ Ⅎ𝑦𝜑
3	2	sbid2 2508	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	sylib 218	1 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1539  Ⅎwnf 1784  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2144  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068

This theorem is referenced by:  sbtr  2516