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Theorem optocl 5757
Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.) Shorten and reduce axiom usage. (Revised by TM, 29-Dec-2025.)
Hypotheses
Ref Expression
optocl.1 𝐷 = (𝐵 × 𝐶)
optocl.2 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
optocl.3 ((𝑥𝐵𝑦𝐶) → 𝜑)
Assertion
Ref Expression
optocl (𝐴𝐷𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐷(𝑥, 𝑦)

Proof of Theorem optocl
StepHypRef Expression
1 elxpi 5685 . . 3 (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)))
2 optocl.3 . . . . . 6 ((𝑥𝐵𝑦𝐶) → 𝜑)
3 optocl.2 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
43eqcoms 2773 . . . . . 6 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
52, 4imbitrid 247 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → ((𝑥𝐵𝑦𝐶) → 𝜓))
65imp 412 . . . 4 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) → 𝜓)
76exlimivv 1965 . . 3 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐵𝑦𝐶)) → 𝜓)
81, 7syl 18 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝜓)
9 optocl.1 . 2 𝐷 = (𝐵 × 𝐶)
108, 9eleq2s 2883 1 (𝐴𝐷𝜓)
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wb 209  wa 401   = wceq 1570  wex 1812  wcel 2146  cop 4597   × cxp 5661
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737
This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-opab 5176  df-xp 5669
This theorem is used by:  2optocl  5759  3optocl  5760  ecoptocl  8811  ax1rid  11161  axcnre  11164
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