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Theorem ax9lem15 29154
 Description: Change free variable without using sp 1716, ax9 1889, or ax10 1884. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ax9lem15.a
ax9lem15.b
ax9lem15.c
Assertion
Ref Expression
ax9lem15
Distinct variable groups:   ,,   ,,

Proof of Theorem ax9lem15
StepHypRef Expression
1 ax9lem15.a . . . 4
2 ax9lem15.c . . . 4
3 hbe1 1705 . . . 4
41ax9lem1 29140 . . . . . . . . 9
5 ax9lem15.b . . . . . . . . . 10
65ax9lem1 29140 . . . . . . . . 9
7 ax-8 1643 . . . . . . . . 9
84, 6, 7syl2im 34 . . . . . . . 8
98com12 27 . . . . . . 7
109con3rr3 128 . . . . . 6
111, 2ax9lem3 29142 . . . . . . . 8
1211con2i 112 . . . . . . 7
13 df-ex 1529 . . . . . . 7
1412, 13sylibr 203 . . . . . 6
1510, 14syl6 29 . . . . 5
16 ax-8 1643 . . . . . . 7
1716con3d 125 . . . . . 6
18 ax-17 1603 . . . . . 6
191, 2, 5, 17, 18ax9lem12 29151 . . . . 5
2015, 19pm2.61d1 151 . . . 4
211, 2, 3, 20ax9lem11 29150 . . 3
22 exnal 1561 . . 3
23 exnal 1561 . . 3
2421, 22, 233imtr3i 256 . 2
2524con4i 122 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4  wal 1527  wex 1528 This theorem is referenced by:  ax9lem16  29155 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-8 1643  ax-6 1703  ax-11 1715 This theorem depends on definitions:  df-bi 177  df-ex 1529
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