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Theorem 3optocl 4361
Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
Hypotheses
Ref Expression
3optocl.1 𝑅 = (𝐷 × 𝐹)
3optocl.2 (⟨x, y⟩ = A → (φψ))
3optocl.3 (⟨z, w⟩ = B → (ψχ))
3optocl.4 (⟨v, u⟩ = 𝐶 → (χθ))
3optocl.5 (((x 𝐷 y 𝐹) (z 𝐷 w 𝐹) (v 𝐷 u 𝐹)) → φ)
Assertion
Ref Expression
3optocl ((A 𝑅 B 𝑅 𝐶 𝑅) → θ)
Distinct variable groups:   x,y,z,w,v,u,A   z,B,w,v,u   v,𝐶,u   x,𝐷,y,z,w,v,u   x,𝐹,y,z,w,v,u   z,𝑅,w,v,u   ψ,x,y   χ,z,w   θ,v,u
Allowed substitution hints:   φ(x,y,z,w,v,u)   ψ(z,w,v,u)   χ(x,y,v,u)   θ(x,y,z,w)   B(x,y)   𝐶(x,y,z,w)   𝑅(x,y)

Proof of Theorem 3optocl
StepHypRef Expression
1 3optocl.1 . . . 4 𝑅 = (𝐷 × 𝐹)
2 3optocl.4 . . . . 5 (⟨v, u⟩ = 𝐶 → (χθ))
32imbi2d 219 . . . 4 (⟨v, u⟩ = 𝐶 → (((A 𝑅 B 𝑅) → χ) ↔ ((A 𝑅 B 𝑅) → θ)))
4 3optocl.2 . . . . . . 7 (⟨x, y⟩ = A → (φψ))
54imbi2d 219 . . . . . 6 (⟨x, y⟩ = A → (((v 𝐷 u 𝐹) → φ) ↔ ((v 𝐷 u 𝐹) → ψ)))
6 3optocl.3 . . . . . . 7 (⟨z, w⟩ = B → (ψχ))
76imbi2d 219 . . . . . 6 (⟨z, w⟩ = B → (((v 𝐷 u 𝐹) → ψ) ↔ ((v 𝐷 u 𝐹) → χ)))
8 3optocl.5 . . . . . . 7 (((x 𝐷 y 𝐹) (z 𝐷 w 𝐹) (v 𝐷 u 𝐹)) → φ)
983expia 1105 . . . . . 6 (((x 𝐷 y 𝐹) (z 𝐷 w 𝐹)) → ((v 𝐷 u 𝐹) → φ))
101, 5, 7, 92optocl 4360 . . . . 5 ((A 𝑅 B 𝑅) → ((v 𝐷 u 𝐹) → χ))
1110com12 27 . . . 4 ((v 𝐷 u 𝐹) → ((A 𝑅 B 𝑅) → χ))
121, 3, 11optocl 4359 . . 3 (𝐶 𝑅 → ((A 𝑅 B 𝑅) → θ))
1312impcom 116 . 2 (((A 𝑅 B 𝑅) 𝐶 𝑅) → θ)
14133impa 1098 1 ((A 𝑅 B 𝑅 𝐶 𝑅) → θ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  cop 3370   × cxp 4286
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-opab 3810  df-xp 4294
This theorem is referenced by:  ecopovtrn  6139  ecopovtrng  6142
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