Theorem sbtrt 2547

Description: Partially closed form of sbtr 2548. Usage of this theorem is discouraged because it depends on ax-13 2404. (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 2102	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
2		sbtrt.nf	. . 3 ⊢ Ⅎ𝑦𝜑
3	2	sbid2 2540	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	sylib 221	1 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 → 𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1813  [wsb 2096

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-12 2213  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-nf 1814  df-sb 2097

This theorem is used by:  sbtr  2548