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Theorem ecoptocl 5996
 Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by set.mm contributors, 23-Jul-1995.)
Hypotheses
Ref Expression
ecoptocl.1 S = ((B × C) / R)
ecoptocl.2 ([x, y]R = A → (φψ))
ecoptocl.3 ((x B y C) → φ)
Assertion
Ref Expression
ecoptocl (A Sψ)
Distinct variable groups:   x,y,A   x,B,y   x,C,y   x,R,y   ψ,x,y
Allowed substitution hints:   φ(x,y)   S(x,y)

Proof of Theorem ecoptocl
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 elqsi 5978 . . 3 (A ((B × C) / R) → z (B × C)A = [z]R)
2 eqid 2353 . . . . 5 (B × C) = (B × C)
3 eceq1 5962 . . . . . . 7 (x, y = z → [x, y]R = [z]R)
43eqeq2d 2364 . . . . . 6 (x, y = z → (A = [x, y]RA = [z]R))
54imbi1d 308 . . . . 5 (x, y = z → ((A = [x, y]Rψ) ↔ (A = [z]Rψ)))
6 ecoptocl.3 . . . . . 6 ((x B y C) → φ)
7 ecoptocl.2 . . . . . . 7 ([x, y]R = A → (φψ))
87eqcoms 2356 . . . . . 6 (A = [x, y]R → (φψ))
96, 8syl5ibcom 211 . . . . 5 ((x B y C) → (A = [x, y]Rψ))
102, 5, 9optocl 4838 . . . 4 (z (B × C) → (A = [z]Rψ))
1110rexlimiv 2732 . . 3 (z (B × C)A = [z]Rψ)
121, 11syl 15 . 2 (A ((B × C) / R) → ψ)
13 ecoptocl.1 . 2 S = ((B × C) / R)
1412, 13eleq2s 2445 1 (A Sψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ⟨cop 4561   × cxp 4770  [cec 5945   / cqs 5946 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-ima 4727  df-xp 4784  df-ec 5947  df-qs 5951 This theorem is referenced by:  2ecoptocl  5997  3ecoptocl  5998
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