Theorem sbcth 3806

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 1792	. 2 ⊢ ∀𝑥𝜑
3		spsbc 3804	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 20	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1535   ∈ wcel 2106  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792

This theorem is referenced by:  iota4an  6545  tfinds2  7885  wunnat  18011  wunnatOLD  18012  catcfuccl  18173  catcfucclOLD  18174  dprdval  20038  opsbc2ie  32504  bj-sbceqgALT  36885  f1omptsnlem  37319  mptsnunlem  37321  topdifinffinlem  37330  relowlpssretop  37347  cdlemk35s  40920  cdlemk39s  40922  cdlemk42  40924  frege92  43945