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Theorem cbvopab2davw 36208
Description: Change the second bound variable in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbvopab2davw.1 ((𝜑𝑦 = 𝑧) → (𝜓𝜒))
Assertion
Ref Expression
cbvopab2davw (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑧⟩ ∣ 𝜒})
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝜓,𝑧   𝜒,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑧)

Proof of Theorem cbvopab2davw
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 opeq2 4881 . . . . . . . 8 (𝑦 = 𝑧 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑧⟩)
21eqeq2d 2744 . . . . . . 7 (𝑦 = 𝑧 → (𝑡 = ⟨𝑥, 𝑦⟩ ↔ 𝑡 = ⟨𝑥, 𝑧⟩))
32adantl 481 . . . . . 6 ((𝜑𝑦 = 𝑧) → (𝑡 = ⟨𝑥, 𝑦⟩ ↔ 𝑡 = ⟨𝑥, 𝑧⟩))
4 cbvopab2davw.1 . . . . . 6 ((𝜑𝑦 = 𝑧) → (𝜓𝜒))
53, 4anbi12d 631 . . . . 5 ((𝜑𝑦 = 𝑧) → ((𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑡 = ⟨𝑥, 𝑧⟩ ∧ 𝜒)))
65cbvexdvaw 2034 . . . 4 (𝜑 → (∃𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑧(𝑡 = ⟨𝑥, 𝑧⟩ ∧ 𝜒)))
76exbidv 1917 . . 3 (𝜑 → (∃𝑥𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥𝑧(𝑡 = ⟨𝑥, 𝑧⟩ ∧ 𝜒)))
87abbidv 2804 . 2 (𝜑 → {𝑡 ∣ ∃𝑥𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑥𝑧(𝑡 = ⟨𝑥, 𝑧⟩ ∧ 𝜒)})
9 df-opab 5212 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥𝑦(𝑡 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
10 df-opab 5212 . 2 {⟨𝑥, 𝑧⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑥𝑧(𝑡 = ⟨𝑥, 𝑧⟩ ∧ 𝜒)}
118, 9, 103eqtr4g 2798 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑧⟩ ∣ 𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wex 1774  {cab 2710  cop 4636  {copab 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5212
This theorem is referenced by: (None)
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