Theorem efp2 15290

Description: Euclid's first postulate. There is a unique line passing through two distinct points.

Hypothesis

Ref	Expression
efp2.1	|- P = U.L

Assertion

Ref	Expression
efp2	|- ((L e. Plig /\ A e. P /\ B e. P) -> (A =/= B -> E!l e. L (A e. l /\ B e. l)))

Distinct variable groups:   A,l   B,l   L,l

Proof of Theorem efp2

Step	Hyp	Ref	Expression
1		efp2.1	. . . 4 |- P = U.L
2	1	efp 15289	. . 3 |- (L e. Plig -> A.a e. P A.b e. P (a =/= b -> E!l e. L (a e. l /\ b e. l)))
3		neeq1 2024	. . . . . . 7 |- (a = A -> (a =/= b <-> A =/= b))
4		eleq1 1957	. . . . . . . . 9 |- (a = A -> (a e. l <-> A e. l))
5	4	anbi1d 679	. . . . . . . 8 |- (a = A -> ((a e. l /\ b e. l) <-> (A e. l /\ b e. l)))
6	5	reubidv 2260	. . . . . . 7 |- (a = A -> (E!l e. L (a e. l /\ b e. l) <-> E!l e. L (A e. l /\ b e. l)))
7	3, 6	imbi12d 688	. . . . . 6 |- (a = A -> ((a =/= b -> E!l e. L (a e. l /\ b e. l)) <-> (A =/= b -> E!l e. L (A e. l /\ b e. l))))
8		neeq2 2025	. . . . . . 7 |- (b = B -> (A =/= b <-> A =/= B))
9		eleq1 1957	. . . . . . . . 9 |- (b = B -> (b e. l <-> B e. l))
10	9	anbi2d 678	. . . . . . . 8 |- (b = B -> ((A e. l /\ b e. l) <-> (A e. l /\ B e. l)))
11	10	reubidv 2260	. . . . . . 7 |- (b = B -> (E!l e. L (A e. l /\ b e. l) <-> E!l e. L (A e. l /\ B e. l)))
12	8, 11	imbi12d 688	. . . . . 6 |- (b = B -> ((A =/= b -> E!l e. L (A e. l /\ b e. l)) <-> (A =/= B -> E!l e. L (A e. l /\ B e. l))))
13	7, 12	rcla42v 2384	. . . . 5 |- ((A e. P /\ B e. P) -> (A.a e. P A.b e. P (a =/= b -> E!l e. L (a e. l /\ b e. l)) -> (A =/= B -> E!l e. L (A e. l /\ B e. l))))
14	13	ex 402	. . . 4 |- (A e. P -> (B e. P -> (A.a e. P A.b e. P (a =/= b -> E!l e. L (a e. l /\ b e. l)) -> (A =/= B -> E!l e. L (A e. l /\ B e. l)))))
15	14	com3r 39	. . 3 |- (A.a e. P A.b e. P (a =/= b -> E!l e. L (a e. l /\ b e. l)) -> (A e. P -> (B e. P -> (A =/= B -> E!l e. L (A e. l /\ B e. l)))))
16	2, 15	syl 12	. 2 |- (L e. Plig -> (A e. P -> (B e. P -> (A =/= B -> E!l e. L (A e. l /\ B e. l)))))
17	16	3imp 1061	1 |- ((L e. Plig /\ A e. P /\ B e. P) -> (A =/= B -> E!l e. L (A e. l /\ B e. l)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E!wreu 2107  U.cuni 3177  Pligcplig 10343

This theorem is referenced by:  isline1 15294

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-uni 3178  df-plig 10344

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