Theorem sbtrt 2534

Description: Partially closed form of sbtr 2535. Usage of this theorem is discouraged because it depends on ax-13 2379. (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 2073	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
2		sbtrt.nf	. . 3 ⊢ Ⅎ𝑦𝜑
3	2	sbid2 2527	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	sylib 221	1 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1536  Ⅎwnf 1785  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070

This theorem is referenced by:  sbtr  2535