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Theorem opabbid 3813
 Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (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 437 . . . . 5 (φ → ((z = ⟨x, y ψ) ↔ (z = ⟨x, y χ)))
52, 4exbid 1504 . . . 4 (φ → (y(z = ⟨x, y ψ) ↔ y(z = ⟨x, y χ)))
61, 5exbid 1504 . . 3 (φ → (xy(z = ⟨x, y ψ) ↔ xy(z = ⟨x, y χ)))
76abbidv 2152 . 2 (φ → {zxy(z = ⟨x, y ψ)} = {zxy(z = ⟨x, y χ)})
8 df-opab 3810 . 2 {⟨x, y⟩ ∣ ψ} = {zxy(z = ⟨x, y ψ)}
9 df-opab 3810 . 2 {⟨x, y⟩ ∣ χ} = {zxy(z = ⟨x, y χ)}
107, 8, 93eqtr4g 2094 1 (φ → {⟨x, y⟩ ∣ ψ} = {⟨x, y⟩ ∣ χ})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  Ⅎwnf 1346  ∃wex 1378  {cab 2023  ⟨cop 3370  {copab 3808 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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-opab 3810 This theorem is referenced by:  opabbidv  3814  mpteq12f  3828  fnoprabg  5544
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