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Theorem opabbid 5107
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Hypotheses
Ref Expression
opabbid.1 𝑥𝜑
opabbid.2 𝑦𝜑
opabbid.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
opabbid (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Proof of Theorem opabbid
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 opabbid.1 . . . 4 𝑥𝜑
2 opabbid.2 . . . . 5 𝑦𝜑
3 opabbid.3 . . . . . 6 (𝜑 → (𝜓𝜒))
43anbi2d 630 . . . . 5 (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
52, 4exbid 2225 . . . 4 (𝜑 → (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
61, 5exbid 2225 . . 3 (𝜑 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)))
76abbidv 2884 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)})
8 df-opab 5105 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
9 df-opab 5105 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)}
107, 8, 93eqtr4g 2880 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wex 1780  wnf 1784  {cab 2798  cop 4549  {copab 5104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-opab 5105
This theorem is referenced by:  opabbidv  5108  mpteq12df  5124  mpteq12f  5125  feqmptdf  6711  fnoprabg  7252  sprsymrelfo  43805
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