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Theorem elabreximd 27082
 Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
elabreximd.1
elabreximd.2
elabreximd.3
elabreximd.4
elabreximd.5
Assertion
Ref Expression
elabreximd
Distinct variable groups:   ,,   ,   ,
Allowed substitution hints:   (,)   (,)   (,)   ()   ()   (,)

Proof of Theorem elabreximd
StepHypRef Expression
1 elabreximd.4 . . . 4
2 eqeq1 2471 . . . . . 6
32rexbidv 2973 . . . . 5
43elabg 3251 . . . 4
51, 4syl 16 . . 3
65biimpa 484 . 2
7 elabreximd.1 . . . 4
8 elabreximd.2 . . . 4
9 simpr 461 . . . . . 6
10 elabreximd.5 . . . . . . 7
1110adantr 465 . . . . . 6
12 elabreximd.3 . . . . . . 7
1312biimpar 485 . . . . . 6
149, 11, 13syl2anc 661 . . . . 5
1514exp31 604 . . . 4
167, 8, 15rexlimd 2947 . . 3
1716imp 429 . 2
186, 17syldan 470 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379  wnf 1599   wcel 1767  cab 2452  wrex 2815 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  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115 This theorem is referenced by:  elabreximdv  27083  abrexss  27084  disjabrex  27116  disjabrexf  27117
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