Theorem sb9 921

Description: Commutation of quantification and substitution variables.

Assertion

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

Proof of Theorem sb9

Step	Hyp	Ref	Expression
1		sb9i 920	. 2 ⊢ (∀x[x / y]φ → ∀y[y / x]φ)
2		sb9i 920	. 2 ⊢ (∀y[y / x]φ → ∀x[x / y]φ)
3	1, 2	impbi 139	1 ⊢ (∀x[x / y]φ ↔ ∀y[y / x]φ)

Syntax hints:   ↔ wb 127  ∀wal 672  [wsb 852

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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