Theorem sb9v 1903

Description: Like sb9 1904 but with a distinct variable constraint between 𝑥 and 𝑦. (Contributed by Jim Kingdon, 28-Feb-2018.)

Assertion

Ref	Expression
sb9v	⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb9v

Step	Hyp	Ref	Expression
1		hbs1 1863	. 2 ⊢ ([𝑥 / 𝑦]𝜑 → ∀𝑦[𝑥 / 𝑦]𝜑)
2		hbs1 1863	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3		sbequ12 1702	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
4	3	equcoms 1642	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
5		sbequ12 1702	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
6	4, 5	bitr3d 189	. 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑))
7	1, 2, 6	cbvalh 1684	1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 104  ∀wal 1288  [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694

This theorem is referenced by:  sb9  1904