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Theorem ralcom2 2677
 Description: Commutation of restricted quantifiers. Note that and needn't be distinct (this makes the proof longer). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
Assertion
Ref Expression
ralcom2
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)

Proof of Theorem ralcom2
StepHypRef Expression
1 eleq1 2316 . . . . . . 7
21a4s 1700 . . . . . 6
32imbi1d 310 . . . . . . . . 9
43dral1 1856 . . . . . . . 8
54bicomd 194 . . . . . . 7
6 df-ral 2521 . . . . . . 7
7 df-ral 2521 . . . . . . 7
85, 6, 73bitr4g 281 . . . . . 6
92, 8imbi12d 313 . . . . 5
109dral1 1856 . . . 4
11 df-ral 2521 . . . 4
12 df-ral 2521 . . . 4
1310, 11, 123bitr4g 281 . . 3
1413biimpd 200 . 2
15 nfnae 1847 . . . . 5
16 nfra2 2570 . . . . 5
1715, 16nfan 1737 . . . 4
18 nfnae 1847 . . . . . . . 8
19 nfra1 2566 . . . . . . . 8
2018, 19nfan 1737 . . . . . . 7
21 nfcvf 2414 . . . . . . . . 9
2221adantr 453 . . . . . . . 8
23 nfcvd 2393 . . . . . . . 8
2422, 23nfeld 2407 . . . . . . 7
2520, 24nfan1 1806 . . . . . 6
26 ra42 2578 . . . . . . . . 9
2726ancomsd 442 . . . . . . . 8
2827expdimp 428 . . . . . . 7
2928adantll 697 . . . . . 6
3025, 29ralrimi 2597 . . . . 5
3130ex 425 . . . 4
3217, 31ralrimi 2597 . . 3
3332ex 425 . 2
3414, 33pm2.61i 158 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178   wa 360  wal 1532   wceq 1619   wcel 1621  wnfc 2379  wral 2516 This theorem is referenced by:  tz7.48lem  6407  tratrb  27336  tratrbVD  27671 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-ext 2237 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2521
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