Theorem sbcth 3765

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 1795	. 2 ⊢ ∀𝑥𝜑
3		spsbc 3763	. 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 2109  [wsbc 3750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-sbc 3751

This theorem is referenced by:  iota4an  6481  tfinds2  7820  wunnat  17897  catcfuccl  18056  dprdval  19911  opsbc2ie  32378  bj-sbceqgALT  36863  f1omptsnlem  37297  mptsnunlem  37299  topdifinffinlem  37308  relowlpssretop  37325  cdlemk35s  40904  cdlemk39s  40906  cdlemk42  40908  frege92  43917