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Theorem antid 5929
 Description: The antisymmetry property. (Contributed by SF, 18-Mar-2015.)
Hypotheses
Ref Expression
antid.1 (φR Antisym A)
antid.2 (φX A)
antid.3 (φY A)
antid.4 (φXRY)
antid.5 (φYRX)
Assertion
Ref Expression
antid (φX = Y)

Proof of Theorem antid
Dummy variables a r x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 antid.4 . 2 (φXRY)
2 antid.5 . 2 (φYRX)
3 antid.1 . . . 4 (φR Antisym A)
4 brex 4689 . . . . . 6 (R Antisym A → (R V A V))
5 breq 4641 . . . . . . . . . 10 (r = R → (xryxRy))
6 breq 4641 . . . . . . . . . 10 (r = R → (yrxyRx))
75, 6anbi12d 691 . . . . . . . . 9 (r = R → ((xry yrx) ↔ (xRy yRx)))
87imbi1d 308 . . . . . . . 8 (r = R → (((xry yrx) → x = y) ↔ ((xRy yRx) → x = y)))
982ralbidv 2656 . . . . . . 7 (r = R → (x a y a ((xry yrx) → x = y) ↔ x a y a ((xRy yRx) → x = y)))
10 raleq 2807 . . . . . . . 8 (a = A → (y a ((xRy yRx) → x = y) ↔ y A ((xRy yRx) → x = y)))
1110raleqbi1dv 2815 . . . . . . 7 (a = A → (x a y a ((xRy yRx) → x = y) ↔ x A y A ((xRy yRx) → x = y)))
12 df-antisym 5901 . . . . . . 7 Antisym = {r, a x a y a ((xry yrx) → x = y)}
139, 11, 12brabg 4706 . . . . . 6 ((R V A V) → (R Antisym Ax A y A ((xRy yRx) → x = y)))
144, 13syl 15 . . . . 5 (R Antisym A → (R Antisym Ax A y A ((xRy yRx) → x = y)))
1514ibi 232 . . . 4 (R Antisym Ax A y A ((xRy yRx) → x = y))
163, 15syl 15 . . 3 (φx A y A ((xRy yRx) → x = y))
17 antid.2 . . . 4 (φX A)
18 antid.3 . . . 4 (φY A)
19 breq1 4642 . . . . . . 7 (x = X → (xRyXRy))
20 breq2 4643 . . . . . . 7 (x = X → (yRxyRX))
2119, 20anbi12d 691 . . . . . 6 (x = X → ((xRy yRx) ↔ (XRy yRX)))
22 eqeq1 2359 . . . . . 6 (x = X → (x = yX = y))
2321, 22imbi12d 311 . . . . 5 (x = X → (((xRy yRx) → x = y) ↔ ((XRy yRX) → X = y)))
24 breq2 4643 . . . . . . 7 (y = Y → (XRyXRY))
25 breq1 4642 . . . . . . 7 (y = Y → (yRXYRX))
2624, 25anbi12d 691 . . . . . 6 (y = Y → ((XRy yRX) ↔ (XRY YRX)))
27 eqeq2 2362 . . . . . 6 (y = Y → (X = yX = Y))
2826, 27imbi12d 311 . . . . 5 (y = Y → (((XRy yRX) → X = y) ↔ ((XRY YRX) → X = Y)))
2923, 28rspc2v 2961 . . . 4 ((X A Y A) → (x A y A ((xRy yRx) → x = y) → ((XRY YRX) → X = Y)))
3017, 18, 29syl2anc 642 . . 3 (φ → (x A y A ((xRy yRx) → x = y) → ((XRY YRX) → X = Y)))
3116, 30mpd 14 . 2 (φ → ((XRY YRX) → X = Y))
321, 2, 31mp2and 660 1 (φX = Y)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   class class class wbr 4639   Antisym cantisym 5890 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-antisym 5901 This theorem is referenced by:  nchoicelem8  6296  nchoicelem19  6307
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