Theorem hbsb 987

Description: If z is not free in φ, it is not free in [y / x]φ when y and z are distinct.

Hypothesis

Ref	Expression
hbsb.1	⊢ (φ → ∀zφ)

Assertion

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

Distinct variable group(s):   y,z

Proof of Theorem hbsb

Step	Hyp	Ref	Expression
1		ax-16 922	. 2 ⊢ (∀z z = y → ([y / x]φ → ∀z[y / x]φ))
2		hbsb.1	. . 3 ⊢ (φ → ∀zφ)
3	2	hbsb4 905	. 2 ⊢ (¬ ∀z z = y → ([y / x]φ → ∀z[y / x]φ))
4	1, 3	pm2.61i 110	1 ⊢ ([y / x]φ → ∀z[y / x]φ)

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

This theorem is referenced by:  opabsb 2114  oprabval4g 3053

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  ax-16 922

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

metamath.org