Theorem sbcth 3060

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 1546	. 2 ⊢ ∀xφ
3		spsbc 3058	. 2 ⊢ (A ∈ V → (∀xφ → [̣A / x]̣φ))
4	2, 3	mpi 16	1 ⊢ (A ∈ V → [̣A / x]̣φ)

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710  [̣wsbc 3046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861  df-sbc 3047

This theorem is referenced by:  iota4an  4358