Theorem sb9i 920

Description: Commutation of quantification and substitution variables.

Assertion

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

Proof of Theorem sb9i

Step	Hyp	Ref	Expression
1		del43 856	. . . 4 ⊢ (∀y y = x → ([x / y]φ → [x / x]φ))
2		del45 879	. . . 4 ⊢ (∀y y = x → ([x / x]φ → [y / x]φ))
3	1, 2	syld 27	. . 3 ⊢ (∀y y = x → ([x / y]φ → [y / x]φ))
4	3	del35 836	. 2 ⊢ (∀y y = x → (∀x[x / y]φ → ∀y[y / x]φ))
5		hbsb2 873	. . . . 5 ⊢ (¬ ∀y y = x → ([x / y]φ → ∀y[x / y]φ))
6	5	19.20ii 692	. . . 4 ⊢ (∀x ¬ ∀y y = x → (∀x[x / y]φ → ∀x∀y[x / y]φ))
7	6	eq6s 827	. . 3 ⊢ (¬ ∀y y = x → (∀x[x / y]φ → ∀x∀y[x / y]φ))
8		stdpc4 869	. . . . . 6 ⊢ (∀x[x / y]φ → [y / x][x / y]φ)
9		sbco 910	. . . . . 6 ⊢ ([y / x][x / y]φ ↔ [y / x]φ)
10	8, 9	sylib 173	. . . . 5 ⊢ (∀x[x / y]φ → [y / x]φ)
11	10	19.20i 691	. . . 4 ⊢ (∀y∀x[x / y]φ → ∀y[y / x]φ)
12	11	a7s 689	. . 3 ⊢ (∀x∀y[x / y]φ → ∀y[y / x]φ)
13	7, 12	syl6 23	. 2 ⊢ (¬ ∀y y = x → (∀x[x / y]φ → ∀y[y / x]φ))
14	4, 13	pm2.61i 110	1 ⊢ (∀x[x / y]φ → ∀y[y / x]φ)

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672   = weq 797  [wsb 852

This theorem is referenced by:  sb9 921

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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