Theorem cla43egv 1866

Description: Existential specialization with 3 quantifiers, using implicit substitution.

Hypothesis

Ref	Expression
cla43egv.1	|- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))

Assertion

Ref	Expression
cla43egv	|- ((A e. R /\ B e. S /\ C e. T) -> (ps -> E.xE.yE.zph))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,C,y,z   ps,x,y,z

Proof of Theorem cla43egv

Step	Hyp	Ref	Expression
1		cla43egv.1	. . . . 5 |- ((x = A /\ y = B /\ z = C) -> (ph <-> ps))
2	1	biimprcd 156	. . . 4 |- (ps -> ((x = A /\ y = B /\ z = C) -> ph))
3	2	19.22dv 1290	. . 3 |- (ps -> (E.z(x = A /\ y = B /\ z = C) -> E.zph))
4	3	19.22dvv 1292	. 2 |- (ps -> (E.xE.yE.z(x = A /\ y = B /\ z = C) -> E.xE.yE.zph))
5		elex 1819	. . . 4 |- (A e. R -> E.x x = A)
6		elex 1819	. . . 4 |- (B e. S -> E.y y = B)
7		elex 1819	. . . 4 |- (C e. T -> E.z z = C)
8	5, 6, 7	3anim123i 821	. . 3 |- ((A e. R /\ B e. S /\ C e. T) -> (E.x x = A /\ E.y y = B /\ E.z z = C))
9		eeeanv 1324	. . 3 |- (E.xE.yE.z(x = A /\ y = B /\ z = C) <-> (E.x x = A /\ E.y y = B /\ E.z z = C))
10	8, 9	sylibr 200	. 2 |- ((A e. R /\ B e. S /\ C e. T) -> E.xE.yE.z(x = A /\ y = B /\ z = C))
11	4, 10	syl5com 52	1 |- ((A e. R /\ B e. S /\ C e. T) -> (ps -> E.xE.yE.zph))

Syntax hints:   -> wi 3   <-> wb 146   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  cla43gv 1867

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-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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