Theorem sbid2h 1902

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 1863	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3	1	sbh 1829	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	2, 3	bitri 184	1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1400  [wsb 1815

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-sb 1816

This theorem is referenced by:  sbid2  1903  sb5rf  1905  sb6rf  1906  sbid2v  2056