Theorem sbcth 3743

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 1797	. 2 ⊢ ∀𝑥𝜑
3		spsbc 3741	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 20	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 2114  [wsbc 3728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-sbc 3729

This theorem is referenced by:  iota4an  6480  tfinds2  7815  wunnat  17926  catcfuccl  18085  dprdval  19980  opsbc2ie  32545  bj-sbceqgALT  37209  f1omptsnlem  37652  mptsnunlem  37654  topdifinffinlem  37663  relowlpssretop  37680  cdlemk35s  41383  cdlemk39s  41385  cdlemk42  41387  frege92  44382