Theorem sbid2 2541

Description: An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2405. Check out sbid2vw 2296 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 2540	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
2		sbid2.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	sbf 2307	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	bitri 277	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  Ⅎwnf 1805  [wsb 2092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-12 2214  ax-13 2405

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1802  df-nf 1806  df-sb 2093

This theorem is referenced by:  sbid2v  2542  sbtrt  2548