Theorem sbid2 1838

Description: An identity law for substitution. (Contributed by NM, 5-Aug-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		sbid2.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1507	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	sbid2h 1837	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 104  Ⅎwnf 1448  [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sbco4lem  1994