Theorem sbcth 3754

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 3752	. 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 2145  [wsbc 3739

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-sbc 3740

This theorem is used by:  iota4an  6515  tfinds2  7861  wunnat  18049  catcfuccl  18208  dprdval  20133  opsbc2ie  32952  bj-sbceqgALT  37646  f1omptsnlem  38091  mptsnunlem  38093  topdifinffinlem  38102  relowlpssretop  38119  cdlemk35s  41811  cdlemk39s  41813  cdlemk42  41815  frege92  44796