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Theorem ralcom2 2878
 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 2502 . . . . . . 7
21sps 1772 . . . . . 6
32imbi1d 310 . . . . . . . . 9
43dral1 2060 . . . . . . . 8
54bicomd 194 . . . . . . 7
6 df-ral 2716 . . . . . . 7
7 df-ral 2716 . . . . . . 7
85, 6, 73bitr4g 281 . . . . . 6
92, 8imbi12d 313 . . . . 5
109dral1 2060 . . . 4
11 df-ral 2716 . . . 4
12 df-ral 2716 . . . 4
1310, 11, 123bitr4g 281 . . 3
1413biimpd 200 . 2
15 nfnae 2047 . . . . 5
16 nfra2 2766 . . . . 5
1715, 16nfan 1848 . . . 4
18 nfnae 2047 . . . . . . . 8
19 nfra1 2762 . . . . . . . 8
2018, 19nfan 1848 . . . . . . 7
21 nfcvf 2600 . . . . . . . . 9
2221adantr 453 . . . . . . . 8
23 nfcvd 2579 . . . . . . . 8
2422, 23nfeld 2593 . . . . . . 7
2520, 24nfan1 1847 . . . . . 6
26 rsp2 2774 . . . . . . . . 9
2726ancomsd 442 . . . . . . . 8
2827expdimp 428 . . . . . . 7
2928adantll 696 . . . . . 6
3025, 29ralrimi 2793 . . . . 5
3130ex 425 . . . 4
3217, 31ralrimi 2793 . . 3
3332ex 425 . 2
3414, 33pm2.61i 159 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360  wal 1550   wcel 1727  wnfc 2565  wral 2711 This theorem is referenced by:  tz7.48lem  6727  tratrb  28718  tratrbVD  29071 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716
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