Theorem sbid2vw 2260

Description: Reverting substitution yields the original expression. Based on fewer axioms than sbid2v 2551, at the expense of an extra distinct variable condition. (Contributed by NM, 14-May-1993.) (Revised by Wolf Lammen, 5-Aug-2023.)

Assertion

Ref	Expression
sbid2vw	⊢ ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑 ↔ 𝜑)

Distinct variable groups:   𝑥,𝑡   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑡)

Proof of Theorem sbid2vw

Step	Hyp	Ref	Expression
1		sbequ12r 2254	. 2 ⊢ (𝑥 = 𝑡 → ([𝑥 / 𝑡]𝜑 ↔ 𝜑))
2	1	sbievw 2103	1 ⊢ ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑 ↔ 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:  sbco4lem  2283  ichid  43660  dfich2bi  43664