Theorem wl-sb9v 37933

Description: Commutation of quantification and substitution variables based on fewer axioms than sb9 2529. (Contributed by Wolf Lammen, 27-Apr-2025.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-sb9v

Step	Hyp	Ref	Expression
1		alcom 2172	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb6 2097	. . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑))
3		equcom 2026	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
4	3	imbi1i 351	. . . . 5 ⊢ ((𝑦 = 𝑥 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
5	4	albii 1827	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
6	2, 5	bitri 277	. . 3 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
7	6	albii 1827	. 2 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑))
8		sb6 2097	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
9	8	albii 1827	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
10	1, 7, 9	3bitr4i 305	1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1546  [wsb 2074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-11 2170

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075

This theorem is referenced by:  wl-sb8ft  37934