Theorem hbsb3 2043

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 1652	. 2 ⊢ ([y / x]φ → [y / x]∀yφ)
3		hbsb2a 2041	. 2 ⊢ ([y / x]∀yφ → ∀x[y / x]φ)
4	2, 3	syl 15	1 ⊢ ([y / x]φ → ∀x[y / x]φ)

Syntax hints:   → 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  nfs1  2044  ax16ALT  2047