Theorem sbcgf 1986

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 1952	. 2 |- (A e. B -> ([A / y][y / x]ph <-> [A / x]ph))
2		sbcgf.1	. . . . 5 |- (ph -> A.xph)
3	2	sbf 1186	. . . 4 |- ([y / x]ph <-> ph)
4	3	sbcbii 1978	. . 3 |- (A e. B -> ([A / y][y / x]ph <-> [A / y]ph))
5		sbc5g 1954	. . 3 |- (A e. B -> ([A / y]ph <-> E.y(y = A /\ ph)))
6		elex 1819	. . . . 5 |- (A e. B -> E.y y = A)
7	6	biantrurd 727	. . . 4 |- (A e. B -> (ph <-> (E.y y = A /\ ph)))
8		19.41v 1305	. . . 4 |- (E.y(y = A /\ ph) <-> (E.y y = A /\ ph))
9	7, 8	syl6rbbr 539	. . 3 |- (A e. B -> (E.y(y = A /\ ph) <-> ph))
10	4, 5, 9	3bitrd 544	. 2 |- (A e. B -> ([A / y][y / x]ph <-> ph))
11	1, 10	bitr3d 530	1 |- (A e. B -> ([A / x]ph <-> ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170

This theorem is referenced by:  sbc19.21g 1987  sbcabel 1996  csbconstgf 2010  sbcel12g 2011  intab 2560  csbopabg 2678  dfoprab5 4115  foprab2 4119  fsumcnlem 7989

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

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