Theorem sbid2 2413

Description: An identity law for substitution. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

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 2412	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
2		sbid2.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	sbf 2380	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	bitri 264	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  Ⅎwnf 1708  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbtrt  2420  sbid2v  2457