Theorem sbco 2527

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2380. See sbcov 2256 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbco

Step	Hyp	Ref	Expression
1		sbcom3 2526	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2		sbid 2255	. . 3 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑)
3	2	sbbii 2082	. 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	1, 3	bitri 278	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 209  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-12 2176  ax-13 2380

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-nf 1787  df-sb 2071

This theorem is referenced by:  sbid2  2528  sbco3  2533