Theorem cncnplem3 7773

Description: Lemma for cncnp2 7776.

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 2066	. . . . . . 7 |- (A (_ X -> (x e. A -> x e. X))
2	1	pm4.71rd 641	. . . . . 6 |- (A (_ X -> (x e. A <-> (x e. X /\ x e. A)))
3	2	imbi1d 615	. . . . 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 538	. . . 4 |- (A (_ X -> ((x e. A -> x e. B) <-> (x e. X -> (x e. A -> x e. B))))
6	5	albidv 1280	. . 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 1652	. . 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 540	. 2 |- (A (_ X -> (A.x(x e. A -> x e. B) <-> A.x e. X (x e. A -> x e. B)))
9		cncnplem2 7772	. 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 956   e. wcel 960  A.wral 1648   (_ wss 2050  U_ciun 2570

This theorem is referenced by:  cncnplem4 7774

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-in 2054  df-ss 2056  df-iun 2572

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