Theorem sbcth 3676

Description: A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)

Hypothesis

Ref	Expression
sbcth.1	⊢ 𝜑

Assertion

Ref	Expression
sbcth	⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑)

Proof of Theorem sbcth

Step	Hyp	Ref	Expression
1		sbcth.1	. . 3 ⊢ 𝜑
2	1	ax-gen 1896	. 2 ⊢ ∀𝑥𝜑
3		spsbc 3674	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 20	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1656   ∈ wcel 2166  [wsbc 3661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-13 2390  ax-ext 2802

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-v 3415  df-sbc 3662

This theorem is referenced by:  iota4an  6104  tfinds2  7323  wunnat  16967  catcfuccl  17110  dprdval  18755  bj-sbceqgALT  33417  f1omptsnlem  33728  mptsnunlem  33730  topdifinffinlem  33739  relowlpssretop  33756  cnfinltrel  33785  cdlemk35s  37011  cdlemk39s  37013  cdlemk42  37015  frege92  39088