Theorem sb9 2095

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

Assertion

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

Proof of Theorem sb9

Step	Hyp	Ref	Expression
1		sb9i 2094	. 2 ⊢ (∀x[x / y]φ → ∀y[y / x]φ)
2		sb9i 2094	. 2 ⊢ (∀y[y / x]φ → ∀x[x / y]φ)
3	1, 2	impbii 180	1 ⊢ (∀x[x / y]φ ↔ ∀y[y / x]φ)

Syntax hints:   ↔ wb 176  ∀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: (None)