Theorem hbsb2 873

Description: Substitution with a distinct variable makes the substituted variable not free.

Assertion

Ref	Expression
hbsb2	⊢ (¬ ∀x x = y → ([y / x]φ → ∀x[y / x]φ))

Proof of Theorem hbsb2

Step	Hyp	Ref	Expression
1		sb4 861	. 2 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))
2		sb2 859	. . 3 ⊢ (∀x(x = y → φ) → [y / x]φ)
3	2	a5i 687	. 2 ⊢ (∀x(x = y → φ) → ∀x[y / x]φ)
4	1, 3	syl6 23	1 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x[y / x]φ))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672   = weq 797  [wsb 852

This theorem is referenced by:  hbsb3 875  sbequi 876  hbsb4 905  sbidm 912  sbco3 915  sb9i 920  hbs1 986

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-an 198  df-ex 679  df-sb 853

metamath.org