Theorem sbcth 1943

Description: A substitution into a theorem remains true (when A is a set).

Hypothesis

Ref	Expression
sbcth.1	⊢ φ

Assertion

Ref	Expression
sbcth	⊢ (A ∈ B → [A / x]φ)

Proof of Theorem sbcth

Step	Hyp	Ref	Expression
1		sbcth.1	. . 3 ⊢ φ
2	1	ax-gen 962	. 2 ⊢ ∀xφ
3		a4sbc 1942	. 2 ⊢ (A ∈ B → (∀xφ → [A / x]φ))
4	2, 3	mpi 44	1 ⊢ (A ∈ B → [A / x]φ)

Syntax hints:   → wi 3  ∀wal 953   ∈ wcel 957  [wsbc 1169

This theorem is referenced by:  sbcth2 1979  csbeq2i 2017

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939

Copyright terms: Public domain