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Theorem ax9lem12 29203
 Description: Lemma for ax9 1954. Similar to spime 1981 with distinct variables, without using sp 1748, ax9 1954, or ax10 1949. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax9lem12.a
ax9lem12.c
ax9lem12.d
ax9lem12.1
ax9lem12.2
Assertion
Ref Expression
ax9lem12
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem ax9lem12
StepHypRef Expression
1 ax9lem12.d . . . 4
2 ax9lem12.a . . . . . 6
3 ax9lem12.c . . . . . 6
4 id 19 . . . . . . 7
54hbth 1552 . . . . . . . 8
62, 3ax9lem7 29198 . . . . . . . . 9
76a1i 10 . . . . . . . 8
8 ax9lem12.2 . . . . . . . . . 10
92, 3, 8ax9lem8 29199 . . . . . . . . 9
109a1i 10 . . . . . . . 8
112, 3, 5, 7, 10ax9lem9 29200 . . . . . . 7
124, 11ax-mp 8 . . . . . 6
132, 3, 12ax9lem8 29199 . . . . 5
14 ax9lem12.1 . . . . . . 7
152, 3ax9lem3 29194 . . . . . . 7
1614, 15nsyli 133 . . . . . 6
1716con3i 127 . . . . 5
1813, 17alrimih 1565 . . . 4
191, 18mt3 171 . . 3
2019con2i 112 . 2
21 df-ex 1542 . 2
2220, 21sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4  wal 1540  wex 1541 This theorem is referenced by:  ax9lem15  29206 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542
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