Theorem sbcth 3756

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 1827	. 2 ⊢ ∀𝑥𝜑
3		spsbc 3754	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 21	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567   ∈ wcel 2146  [wsbc 3741

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-sbc 3742

This theorem is used by:  iota4an  6527  tfinds2  7882  wunnat  18146  catcfuccl  18305  dprdval  20231  opsbc2ie  33126  bj-sbceqgALT  37941  f1omptsnlem  38399  mptsnunlem  38401  topdifinffinlem  38410  relowlpssretop  38427  cdlemk35s  42134  cdlemk39s  42136  cdlemk42  42138  frege92  45095