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Theorem releqel 5807
Description: Lemma to turn a membership condition into an equality condition. (Contributed by SF, 9-Mar-2015.)
Hypotheses
Ref Expression
releqel.1 T V
releqel.2 ({y}, T Ry A)
Assertion
Ref Expression
releqel (x, T ∼ (( Ins3 S Ins2 R) “ 1c) ↔ x = A)
Distinct variable groups:   y,A   y,R   y,T   x,y
Allowed substitution hints:   A(x)   R(x)   T(x)

Proof of Theorem releqel
StepHypRef Expression
1 elima1c 4947 . . . 4 (x, T (( Ins3 S Ins2 R) “ 1c) ↔ y{y}, x, T ( Ins3 S Ins2 R))
2 elsymdif 3223 . . . . . 6 ({y}, x, T ( Ins3 S Ins2 R) ↔ ¬ ({y}, x, T Ins3 S {y}, x, T Ins2 R))
3 releqel.1 . . . . . . . . 9 T V
43otelins3 5792 . . . . . . . 8 ({y}, x, T Ins3 S {y}, x S )
5 vex 2862 . . . . . . . . 9 y V
6 vex 2862 . . . . . . . . 9 x V
75, 6opelssetsn 4760 . . . . . . . 8 ({y}, x S y x)
84, 7bitri 240 . . . . . . 7 ({y}, x, T Ins3 S y x)
96otelins2 5791 . . . . . . . 8 ({y}, x, T Ins2 R{y}, T R)
10 releqel.2 . . . . . . . 8 ({y}, T Ry A)
119, 10bitri 240 . . . . . . 7 ({y}, x, T Ins2 Ry A)
128, 11bibi12i 306 . . . . . 6 (({y}, x, T Ins3 S {y}, x, T Ins2 R) ↔ (y xy A))
132, 12xchbinx 301 . . . . 5 ({y}, x, T ( Ins3 S Ins2 R) ↔ ¬ (y xy A))
1413exbii 1582 . . . 4 (y{y}, x, T ( Ins3 S Ins2 R) ↔ y ¬ (y xy A))
15 exnal 1574 . . . 4 (y ¬ (y xy A) ↔ ¬ y(y xy A))
161, 14, 153bitrri 263 . . 3 y(y xy A) ↔ x, T (( Ins3 S Ins2 R) “ 1c))
1716con1bii 321 . 2 x, T (( Ins3 S Ins2 R) “ 1c) ↔ y(y xy A))
186, 3opex 4588 . . 3 x, T V
1918elcompl 3225 . 2 (x, T ∼ (( Ins3 S Ins2 R) “ 1c) ↔ ¬ x, T (( Ins3 S Ins2 R) “ 1c))
20 dfcleq 2347 . 2 (x = Ay(y xy A))
2117, 19, 203bitr4i 268 1 (x, T ∼ (( Ins3 S Ins2 R) “ 1c) ↔ x = A)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 176  wal 1540  wex 1541   = wceq 1642   wcel 1710  Vcvv 2859  ccompl 3205  csymdif 3209  {csn 3737  1cc1c 4134  cop 4561   S csset 4719  cima 4722   Ins2 cins2 5749   Ins3 cins3 5751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-sset 4725  df-co 4726  df-ima 4727  df-cnv 4785  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752
This theorem is referenced by:  releqmpt  5808  ceex  6174  nmembers1lem1  6268  nchoicelem10  6298
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