Theorem hbsb3 1686

Description: If y is not free in φ, x is not free in [y / x]φ. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbsb3.1	⊢ (φ → ∀yφ)

Assertion

Ref	Expression
hbsb3	⊢ ([y / x]φ → ∀x[y / x]φ)

Proof of Theorem hbsb3

Step	Hyp	Ref	Expression
1		hbsb3.1	. . 3 ⊢ (φ → ∀yφ)
2	1	sbimi 1644	. 2 ⊢ ([y / x]φ → [y / x]∀yφ)
3		hbsb2a 1684	. 2 ⊢ ([y / x]∀yφ → ∀x[y / x]φ)
4	2, 3	syl 14	1 ⊢ ([y / x]φ → ∀x[y / x]φ)

Syntax hints:   → wi 4  ∀wal 1240  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-11 1394  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by:  nfs1  1687  sbcof2  1688  ax16  1691  sb8h  1731  sb8eh  1732  ax16ALT  1736