Theorem sbcov 2258

Description: A composition law for substitution. Version of sbco 2549 with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Gino Giotto, 7-Aug-2023.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbcov

Step	Hyp	Ref	Expression
1		sbcom3vv 2106	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2		sbid 2257	. . 3 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑)
3	2	sbbii 2081	. 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	1, 3	bitri 277	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 208  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070

This theorem is referenced by:  sb6a  2259  sbbibOLD  2289