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Theorem iserd 5942
 Description: A symmetric, transitive relationship is an equivalence relationship. (Contributed by SF, 22-Feb-2015.)
Hypotheses
Ref Expression
iserd.1 (φR V)
iserd.2 (φA W)
iserd.3 ((φ (x A y A) xRy) → yRx)
iserd.4 ((φ (x A y A z A) (xRy yRz)) → xRz)
Assertion
Ref Expression
iserd (φR Er A)
Distinct variable groups:   x,A,y,z   φ,x,y,z   x,R,y,z
Allowed substitution hints:   V(x,y,z)   W(x,y,z)

Proof of Theorem iserd
Dummy variables a r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iserd.3 . . . . 5 ((φ (x A y A) xRy) → yRx)
213expia 1153 . . . 4 ((φ (x A y A)) → (xRyyRx))
32ralrimivva 2706 . . 3 (φx A y A (xRyyRx))
4 iserd.1 . . . 4 (φR V)
5 iserd.2 . . . 4 (φA W)
6 breq 4641 . . . . . . 7 (r = R → (xryxRy))
7 breq 4641 . . . . . . 7 (r = R → (yrxyRx))
86, 7imbi12d 311 . . . . . 6 (r = R → ((xryyrx) ↔ (xRyyRx)))
982ralbidv 2656 . . . . 5 (r = R → (x a y a (xryyrx) ↔ x a y a (xRyyRx)))
10 raleq 2807 . . . . . 6 (a = A → (y a (xRyyRx) ↔ y A (xRyyRx)))
1110raleqbi1dv 2815 . . . . 5 (a = A → (x a y a (xRyyRx) ↔ x A y A (xRyyRx)))
12 df-sym 5908 . . . . 5 Sym = {r, a x a y a (xryyrx)}
139, 11, 12brabg 4706 . . . 4 ((R V A W) → (R Sym Ax A y A (xRyyRx)))
144, 5, 13syl2anc 642 . . 3 (φ → (R Sym Ax A y A (xRyyRx)))
153, 14mpbird 223 . 2 (φR Sym A)
16 iserd.4 . . 3 ((φ (x A y A z A) (xRy yRz)) → xRz)
174, 5, 16trrd 5925 . 2 (φR Trans A)
18 ersymtr 5932 . 2 (R Er A ↔ (R Sym A R Trans A))
1915, 17, 18sylanbrc 645 1 (φR Er A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2614   class class class wbr 4639   Trans ctrans 5888   Sym csym 5897   Er cer 5898 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-trans 5899  df-sym 5908  df-er 5909 This theorem is referenced by:  ider  5943  eqer  5961  ener  6039
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