Theorem sb9 1967

Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)

Assertion

Ref	Expression
sb9	⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb9

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sb9v 1966	. . 3 ⊢ (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑤[𝑤 / 𝑦][𝑤 / 𝑥]𝜑)
2		sbcom 1963	. . . 4 ⊢ ([𝑤 / 𝑦][𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥][𝑤 / 𝑦]𝜑)
3	2	albii 1458	. . 3 ⊢ (∀𝑤[𝑤 / 𝑦][𝑤 / 𝑥]𝜑 ↔ ∀𝑤[𝑤 / 𝑥][𝑤 / 𝑦]𝜑)
4		sb9v 1966	. . 3 ⊢ (∀𝑤[𝑤 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑)
5	1, 3, 4	3bitri 205	. 2 ⊢ (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑)
6		ax-17 1514	. . . 4 ⊢ (𝜑 → ∀𝑤𝜑)
7	6	sbco2h 1952	. . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8	7	albii 1458	. 2 ⊢ (∀𝑦[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
9	6	sbco2h 1952	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑)
10	9	albii 1458	. 2 ⊢ (∀𝑥[𝑥 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∀𝑥[𝑥 / 𝑦]𝜑)
11	5, 8, 10	3bitr3ri 210	1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 104  ∀wal 1341  [wsb 1750

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sb9i  1968