Theorem sbcovOLD 2292

Description: Obsolete version of sbcov 2291 as of 26-Aug-2025. (Contributed by NM, 14-May-1993.) (Revised by GG, 7-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem sbcovOLD

Step	Hyp	Ref	Expression
1		sbcom3vv 2131	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑦]𝜑)
2		sbid 2290	. . 3 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑)
3	2	sbbii 2109	. 2 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	1, 3	bitri 278	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096

This theorem is used by: (None)