Theorem sbid2h 1859

Description: An identity law for substitution. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
sbid2h.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
sbid2h	⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Proof of Theorem sbid2h

Step	Hyp	Ref	Expression
1		sbid2h.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	sbcof2 1820	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1	sbh 1786	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1361  [wsb 1772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544

This theorem depends on definitions:  df-bi 117  df-sb 1773

This theorem is referenced by:  sbid2  1860  sb5rf  1862  sb6rf  1863  sbid2v  2006