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Theorem ralcom2 2704
 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 2343 . . . . . . 7
21sps 1739 . . . . . 6
32imbi1d 308 . . . . . . . . 9
43dral1 1905 . . . . . . . 8
54bicomd 192 . . . . . . 7
6 df-ral 2548 . . . . . . 7
7 df-ral 2548 . . . . . . 7
85, 6, 73bitr4g 279 . . . . . 6
92, 8imbi12d 311 . . . . 5
109dral1 1905 . . . 4
11 df-ral 2548 . . . 4
12 df-ral 2548 . . . 4
1310, 11, 123bitr4g 279 . . 3
1413biimpd 198 . 2
15 nfnae 1896 . . . . 5
16 nfra2 2597 . . . . 5
1715, 16nfan 1771 . . . 4
18 nfnae 1896 . . . . . . . 8
19 nfra1 2593 . . . . . . . 8
2018, 19nfan 1771 . . . . . . 7
21 nfcvf 2441 . . . . . . . . 9
2221adantr 451 . . . . . . . 8
23 nfcvd 2420 . . . . . . . 8
2422, 23nfeld 2434 . . . . . . 7
2520, 24nfan1 1822 . . . . . 6
26 rsp2 2605 . . . . . . . . 9
2726ancomsd 440 . . . . . . . 8
2827expdimp 426 . . . . . . 7
2928adantll 694 . . . . . 6
3025, 29ralrimi 2624 . . . . 5
3130ex 423 . . . 4
3217, 31ralrimi 2624 . . 3
3332ex 423 . 2
3414, 33pm2.61i 156 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1527   wceq 1623   wcel 1684  wnfc 2406  wral 2543 This theorem is referenced by:  tz7.48lem  6453  tratrb  28299  tratrbVD  28637 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-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548
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