Theorem sbcth 3792

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 1796	. 2 ⊢ ∀𝑥𝜑
3		spsbc 3790	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 20	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1538   ∈ wcel 2105  [wsbc 3777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-sbc 3778

This theorem is referenced by:  iota4an  6525  tfinds2  7857  wunnat  17914  wunnatOLD  17915  catcfuccl  18076  catcfucclOLD  18077  dprdval  19918  opsbc2ie  31998  bj-sbceqgALT  36098  f1omptsnlem  36533  mptsnunlem  36535  topdifinffinlem  36544  relowlpssretop  36561  cdlemk35s  40124  cdlemk39s  40126  cdlemk42  40128  frege92  43021