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Theorem 2ax6elem 2179
 Description: We can always find values matching and , as long as they are represented by distinct variables. This theorem merges two ax6e 1971 instances and into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 32429. (Contributed by Wolf Lammen, 27-Sep-2018.)
Assertion
Ref Expression
2ax6elem

Proof of Theorem 2ax6elem
StepHypRef Expression
1 ax6e 1971 . . . 4
2 nfnae 2031 . . . . . 6
3 nfnae 2031 . . . . . 6
42, 3nfan 1875 . . . . 5
5 nfeqf 2018 . . . . . 6
6 nfr 1821 . . . . . . 7
7 ax6e 1971 . . . . . . . . 9
8 pm3.21 448 . . . . . . . . 9
97, 8eximii 1637 . . . . . . . 8
10919.35i 1666 . . . . . . 7
116, 10syl6 33 . . . . . 6
125, 11syl 16 . . . . 5
134, 12eximd 1830 . . . 4
141, 13mpi 17 . . 3
1514ex 434 . 2
16 ax6e 1971 . . 3
17 nfae 2029 . . . 4
18 equvini 2060 . . . . 5
19 equtrr 1746 . . . . . . 7
2019anim1d 564 . . . . . 6
2120aleximi 1632 . . . . 5
2218, 21syl5 32 . . . 4
2317, 22eximd 1830 . . 3
2416, 23mpi 17 . 2
2515, 24pm2.61d2 160 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369  wal 1377  wex 1596  wnf 1599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968 This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600 This theorem is referenced by:  2ax6e  2180
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