Theorem sb8 2555

Description: Substitution of variable in universal quantifier. Usage of this theorem is discouraged because it depends on ax-13 2410. For a version requiring disjoint variables, but fewer axioms, see sb8f 2392. (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 2526	. 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		sbequ12 2293	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	1, 2, 3	cbval 2436	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 209  ∀wal 1565  Ⅎwnf 1810  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098

This theorem is referenced by:  sbhb  2559  sb8iota  6504  wl-sb8eut  38121