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Theorem dfif3 3955
Description: Alternate definition of the conditional operator df-if 3942. Note that is independent of i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
Hypothesis
Ref Expression
dfif3.1
Assertion
Ref Expression
dfif3
Distinct variable group:   ,

Proof of Theorem dfif3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfif6 3944 . 2
2 dfif3.1 . . . . . 6
3 biidd 237 . . . . . . 7
43cbvabv 2600 . . . . . 6
52, 4eqtri 2486 . . . . 5
65ineq2i 3696 . . . 4
7 dfrab3 3772 . . . 4
86, 7eqtr4i 2489 . . 3
9 dfrab3 3772 . . . 4
10 notab 3767 . . . . . 6
115difeq2i 3618 . . . . . 6
1210, 11eqtr4i 2489 . . . . 5
1312ineq2i 3696 . . . 4
149, 13eqtr2i 2487 . . 3
158, 14uneq12i 3655 . 2
161, 15eqtr4i 2489 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  =wceq 1395  {cab 2442  {crab 2811   cvv 3109  \cdif 3472  u.cun 3473  i^icin 3474  ifcif 3941
This theorem is referenced by:  dfif4  3956
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-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-if 3942
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