Theorem sbcgf 1982

Description: Substitution for a variable not free in a wff does not affect it.

Hypothesis

Ref	Expression
sbcgf.1	|- (ph -> A.xph)

Assertion

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

Proof of Theorem sbcgf

Step	Hyp	Ref	Expression
1		sbccog 1948	. 2 |- (A e. B -> ([A / y][y / x]ph <-> [A / x]ph))
2		sbcgf.1	. . . . 5 |- (ph -> A.xph)
3	2	sbf 1184	. . . 4 |- ([y / x]ph <-> ph)
4	3	sbcbii 1974	. . 3 |- (A e. B -> ([A / y][y / x]ph <-> [A / y]ph))
5		sbc5g 1950	. . 3 |- (A e. B -> ([A / y]ph <-> E.y(y = A /\ ph)))
6		elex 1815	. . . . 5 |- (A e. B -> E.y y = A)
7	6	biantrurd 726	. . . 4 |- (A e. B -> (ph <-> (E.y y = A /\ ph)))
8		19.41v 1303	. . . 4 |- (E.y(y = A /\ ph) <-> (E.y y = A /\ ph))
9	7, 8	syl6rbbr 538	. . 3 |- (A e. B -> (E.y(y = A /\ ph) <-> ph))
10	4, 5, 9	3bitrd 543	. 2 |- (A e. B -> ([A / y][y / x]ph <-> ph))
11	1, 10	bitr3d 529	1 |- (A e. B -> ([A / x]ph <-> ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  [wsbc 1168

This theorem is referenced by:  sbc19.21g 1983  sbcabel 1992  csbconstgf 2006  sbcel12g 2007  intab 2555  csbopabg 2673  dfoprab5 4105  foprab2 4109  fsumcnlem 7939

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

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