Theorem sbcth 3757

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 1828	. 2 ⊢ ∀𝑥𝜑
3		spsbc 3755	. 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 1568   ∈ wcel 2147  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sbc 3743

This theorem is used by:  iota4an  6547  tfinds2  7909  wunnat  18184  catcfuccl  18343  dprdval  20270  opsbc2ie  33171  bj-sbceqgALT  38029  f1omptsnlem  38527  mptsnunlem  38529  topdifinffinlem  38538  relowlpssretop  38555  cdlemk35s  42260  cdlemk39s  42262  cdlemk42  42264  frege92  45216