Theorem wl-sb9v 38314

Description: Commutation of quantification and substitution variables based on fewer axioms than sb9 2550. (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 2196	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb6 2122	. . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑))
3		equcom 2051	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
4	3	imbi1i 352	. . . . 5 ⊢ ((𝑦 = 𝑥 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
5	4	albii 1852	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
6	2, 5	bitri 278	. . 3 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
7	6	albii 1852	. 2 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑))
8		sb6 2122	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
9	8	albii 1852	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
10	1, 7, 9	3bitr4i 306	1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-11 2194

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  wl-sb8ft  38315