Theorem sbcth2 1979

Description: A substitution into a theorem.

Hypothesis

Ref	Expression
sbcth2.1	|- (x e. B -> ph)

Assertion

Ref	Expression
sbcth2	|- (A e. B -> [A / x]ph)

Distinct variable group:   x,B

Proof of Theorem sbcth2

Step	Hyp	Ref	Expression
1		sbcth2.1	. . . 4 |- (x e. B -> ph)
2	1	sbcth 1943	. . 3 |- (A e. B -> [A / x](x e. B -> ph))
3		sbcimg 1967	. . . 4 |- (A e. B -> ([A / x](x e. B -> ph) <-> ([A / x]x e. B -> [A / x]ph)))
4		sbcel1gv 1977	. . . . 5 |- (A e. B -> ([A / x]x e. B <-> A e. B))
5	4	imbi1d 612	. . . 4 |- (A e. B -> (([A / x]x e. B -> [A / x]ph) <-> (A e. B -> [A / x]ph)))
6	3, 5	bitrd 527	. . 3 |- (A e. B -> ([A / x](x e. B -> ph) <-> (A e. B -> [A / x]ph)))
7	2, 6	mpbid 195	. 2 |- (A e. B -> (A e. B -> [A / x]ph))
8	7	pm2.43i 64	1 |- (A e. B -> [A / x]ph)

Syntax hints:   -> wi 3   e. wcel 957  [wsbc 1169

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939

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