Theorem sbid2h 1821

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 1782	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1	sbh 1749	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	2, 3	bitri 183	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  [wsb 1735

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736

This theorem is referenced by:  sbid2  1822  sb5rf  1824  sb6rf  1825  sbid2v  1969