Theorem sb8 1733

Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Hypothesis

Ref	Expression
sb8e.1	⊢ Ⅎyφ

Assertion

Ref	Expression
sb8	⊢ (∀xφ ↔ ∀y[y / x]φ)

Proof of Theorem sb8

Step	Hyp	Ref	Expression
1		sb8e.1	. 2 ⊢ Ⅎyφ
2	1	nfs1 1687	. 2 ⊢ Ⅎx[y / x]φ
3		sbequ12 1651	. 2 ⊢ (x = y → (φ ↔ [y / x]φ))
4	1, 2, 3	cbval 1634	1 ⊢ (∀xφ ↔ ∀y[y / x]φ)

Syntax hints:   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346  [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-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  sbnf2  1854  sb8eu  1910  nfraldya  2352  rabeq0  3241  abeq0  3242  sb8iota  4817