Theorem sb8 2511

Description: Substitution of variable in universal quantifier. Usage of this theorem is discouraged because it depends on ax-13 2366. For a version requiring disjoint variables, but fewer axioms, see sb8f 2344. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb8.1	⊢ Ⅎ𝑦𝜑

Assertion

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

Proof of Theorem sb8

Step	Hyp	Ref	Expression
1		sb8.1	. 2 ⊢ Ⅎ𝑦𝜑
2	1	nfs1 2482	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2236	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbval 2392	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1532  Ⅎwnf 1778  [wsb 2060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2164  ax-13 2366

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775  df-nf 1779  df-sb 2061

This theorem is referenced by:  sbhb  2515  sb8iota  6506  wl-sb8eut  37034