Theorem sb8 2556

Description: Substitution of variable in universal quantifier. For a version requiring disjoint variables, but fewer axioms, see sb8v 2376. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

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	. 2 ⊢ Ⅎ𝑦𝜑
2	1	nfs1 2496	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2286	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbval 2423	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 198  ∀wal 1654  Ⅎwnf 1882  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068

This theorem is referenced by:  sbhb  2573  sbnf2OLD  2574  abv  3423  sb8iota  6093  mo5f  29868  ax11-pm2  33340  bj-nfcf  33436  wl-sb8eut  33896  sbcalf  34451