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Theorem ralcom2 3022
 Description: Commutation of restricted universal quantifiers. Note that and need not 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:   ,   ,

Proof of Theorem ralcom2
StepHypRef Expression
1 eleq1 2529 . . . . . . 7
21sps 1865 . . . . . 6
32imbi1d 317 . . . . . . . . 9
43dral1 2067 . . . . . . . 8
54bicomd 201 . . . . . . 7
6 df-ral 2812 . . . . . . 7
7 df-ral 2812 . . . . . . 7
85, 6, 73bitr4g 288 . . . . . 6
92, 8imbi12d 320 . . . . 5
109dral1 2067 . . . 4
11 df-ral 2812 . . . 4
12 df-ral 2812 . . . 4
1310, 11, 123bitr4g 288 . . 3
1413biimpd 207 . 2
15 nfnae 2058 . . . . 5
16 nfra2 2844 . . . . 5
1715, 16nfan 1928 . . . 4
18 nfnae 2058 . . . . . . . 8
19 nfra1 2838 . . . . . . . 8
2018, 19nfan 1928 . . . . . . 7
21 nfcvf 2644 . . . . . . . . 9
2221adantr 465 . . . . . . . 8
23 nfcvd 2620 . . . . . . . 8
2422, 23nfeld 2627 . . . . . . 7
2520, 24nfan1 1927 . . . . . 6
26 rsp2 2831 . . . . . . . . 9
2726ancomsd 454 . . . . . . . 8
2827expdimp 437 . . . . . . 7
2928adantll 713 . . . . . 6
3025, 29ralrimi 2857 . . . . 5
3130ex 434 . . . 4
3217, 31ralrimi 2857 . . 3
3332ex 434 . 2
3414, 33pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  e.wcel 1818  F/_wnfc 2605  A.wral 2807 This theorem is referenced by:  tz7.48lem  7125  tratrb  33307  tratrbVD  33661  imo72b2  37993 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812
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