Theorem sbcth 3788

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 1789	. 2 ⊢ ∀𝑥𝜑
3		spsbc 3786	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 20	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1531   ∈ wcel 2098  [wsbc 3773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-sbc 3774

This theorem is referenced by:  iota4an  6531  tfinds2  7869  wunnat  17949  wunnatOLD  17950  catcfuccl  18111  catcfucclOLD  18112  dprdval  19972  opsbc2ie  32352  bj-sbceqgALT  36511  f1omptsnlem  36946  mptsnunlem  36948  topdifinffinlem  36957  relowlpssretop  36974  cdlemk35s  40540  cdlemk39s  40542  cdlemk42  40544  frege92  43527