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Theorem ecoptocl 6129
Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 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 6094 . . 3 (A ((B × 𝐶) / 𝑅) → z (B × 𝐶)A = [z]𝑅)
2 eqid 2037 . . . . 5 (B × 𝐶) = (B × 𝐶)
3 eceq1 6077 . . . . . . 7 (⟨x, y⟩ = z → [⟨x, y⟩]𝑅 = [z]𝑅)
43eqeq2d 2048 . . . . . 6 (⟨x, y⟩ = z → (A = [⟨x, y⟩]𝑅A = [z]𝑅))
54imbi1d 220 . . . . 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 2040 . . . . . 6 (A = [⟨x, y⟩]𝑅 → (φψ))
96, 8syl5ibcom 144 . . . . 5 ((x B y 𝐶) → (A = [⟨x, y⟩]𝑅ψ))
102, 5, 9optocl 4359 . . . 4 (z (B × 𝐶) → (A = [z]𝑅ψ))
1110rexlimiv 2421 . . 3 (z (B × 𝐶)A = [z]𝑅ψ)
121, 11syl 14 . 2 (A ((B × 𝐶) / 𝑅) → ψ)
13 ecoptocl.1 . 2 𝑆 = ((B × 𝐶) / 𝑅)
1412, 13eleq2s 2129 1 (A 𝑆ψ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wrex 2301  cop 3370   × cxp 4286  [cec 6040   / cqs 6041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-ec 6044  df-qs 6048
This theorem is referenced by:  2ecoptocl  6130  3ecoptocl  6131  mulidnq  6373  recexnq  6374  ltsonq  6382  distrnq0  6441  addassnq0  6444  ltposr  6671  0idsr  6675  1idsr  6676  00sr  6677  recexgt0sr  6681  archsr  6688
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