Theorem cncnplem3 7776

Description: Lemma for cncnp2 7779.

Assertion

Ref	Expression
cncnplem3	|- (A (_ X -> (A.x e. X (x e. A -> x e. B) -> A (_ U_x e. A B))

Distinct variable groups:   x,A   x,X

Proof of Theorem cncnplem3

Step	Hyp	Ref	Expression
1		ssel 2063	. . . . . . 7 |- (A (_ X -> (x e. A -> x e. X))
2	1	pm4.71rd 639	. . . . . 6 |- (A (_ X -> (x e. A <-> (x e. X /\ x e. A)))
3	2	imbi1d 613	. . . . 5 |- (A (_ X -> ((x e. A -> x e. B) <-> ((x e. X /\ x e. A) -> x e. B)))
4		impexp 347	. . . . 5 |- (((x e. X /\ x e. A) -> x e. B) <-> (x e. X -> (x e. A -> x e. B)))
5	3, 4	syl6bb 536	. . . 4 |- (A (_ X -> ((x e. A -> x e. B) <-> (x e. X -> (x e. A -> x e. B))))
6	5	albidv 1278	. . 3 |- (A (_ X -> (A.x(x e. A -> x e. B) <-> A.x(x e. X -> (x e. A -> x e. B))))
7		df-ral 1649	. . 3 |- (A.x e. X (x e. A -> x e. B) <-> A.x(x e. X -> (x e. A -> x e. B)))
8	6, 7	syl6bbr 538	. 2 |- (A (_ X -> (A.x(x e. A -> x e. B) <-> A.x e. X (x e. A -> x e. B)))
9		cncnplem2 7775	. 2 |- (A.x(x e. A -> x e. B) -> A (_ U_x e. A B)
10	8, 9	syl6bir 215	1 |- (A (_ X -> (A.x e. X (x e. A -> x e. B) -> A (_ U_x e. A B))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  A.wral 1645   (_ wss 2047  U_ciun 2566

This theorem is referenced by:  cncnplem4 7777

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-10 966  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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-in 2051  df-ss 2053  df-iun 2568

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