Theorem sbid2h 1729

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 1691	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1	sbh 1659	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	2, 3	bitri 173	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  sbid2  1730  sb5rf  1732  sb6rf  1733  sbid2v  1872