Theorem sbid2 2540

Description: An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2404. Check out sbid2vw 2295 for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sbid2.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
sbid2	⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Proof of Theorem sbid2

Step	Hyp	Ref	Expression
1		sbco 2539	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
2		sbid2.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	sbf 2306	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	bitri 278	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  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:  sbid2v  2541  sbtrt  2547