Theorem sb9i 2094

Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sb9i	⊢ (∀x[x / y]φ → ∀y[y / x]φ)

Proof of Theorem sb9i

Step	Hyp	Ref	Expression
1		drsb1 2022	. . . . 5 ⊢ (∀y y = x → ([y / y]φ ↔ [y / x]φ))
2		drsb2 2061	. . . . 5 ⊢ (∀y y = x → ([y / y]φ ↔ [x / y]φ))
3	1, 2	bitr3d 246	. . . 4 ⊢ (∀y y = x → ([y / x]φ ↔ [x / y]φ))
4	3	dral1 1965	. . 3 ⊢ (∀y y = x → (∀y[y / x]φ ↔ ∀x[x / y]φ))
5	4	biimprd 214	. 2 ⊢ (∀y y = x → (∀x[x / y]φ → ∀y[y / x]φ))
6		nfnae 1956	. . . 4 ⊢ Ⅎx ¬ ∀y y = x
7		hbsb2 2057	. . . 4 ⊢ (¬ ∀y y = x → ([x / y]φ → ∀y[x / y]φ))
8	6, 7	alimd 1764	. . 3 ⊢ (¬ ∀y y = x → (∀x[x / y]φ → ∀x∀y[x / y]φ))
9		stdpc4 2024	. . . . . 6 ⊢ (∀x[x / y]φ → [y / x][x / y]φ)
10		sbco 2083	. . . . . 6 ⊢ ([y / x][x / y]φ ↔ [y / x]φ)
11	9, 10	sylib 188	. . . . 5 ⊢ (∀x[x / y]φ → [y / x]φ)
12	11	alimi 1559	. . . 4 ⊢ (∀y∀x[x / y]φ → ∀y[y / x]φ)
13	12	a7s 1735	. . 3 ⊢ (∀x∀y[x / y]φ → ∀y[y / x]φ)
14	8, 13	syl6 29	. 2 ⊢ (¬ ∀y y = x → (∀x[x / y]φ → ∀y[y / x]φ))
15	5, 14	pm2.61i 156	1 ⊢ (∀x[x / y]φ → ∀y[y / x]φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sb9  2095