Theorem sb8 2092

Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb5rf.1	⊢ Ⅎyφ

Assertion

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

Proof of Theorem sb8

Step	Hyp	Ref	Expression
1		sb5rf.1	. . . 4 ⊢ Ⅎyφ
2	1	nfal 1842	. . 3 ⊢ Ⅎy∀xφ
3		stdpc4 2024	. . 3 ⊢ (∀xφ → [y / x]φ)
4	2, 3	alrimi 1765	. 2 ⊢ (∀xφ → ∀y[y / x]φ)
5	1	nfs1 2044	. . 3 ⊢ Ⅎx[y / x]φ
6		stdpc7 1917	. . 3 ⊢ (y = x → ([y / x]φ → φ))
7	5, 1, 6	cbv3 1982	. 2 ⊢ (∀y[y / x]φ → ∀xφ)
8	4, 7	impbii 180	1 ⊢ (∀xφ ↔ ∀y[y / x]φ)

Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544  [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:  sb8e  2093  sbhb  2107  sbnf2  2108  sb8eu  2222  sb8iota  4346