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Theorem opabbid 4624
 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 684 . . . . 5 (φ → ((z = x, y ψ) ↔ (z = x, y χ)))
52, 4exbid 1773 . . . 4 (φ → (y(z = x, y ψ) ↔ y(z = x, y χ)))
61, 5exbid 1773 . . 3 (φ → (xy(z = x, y ψ) ↔ xy(z = x, y χ)))
76abbidv 2467 . 2 (φ → {z xy(z = x, y ψ)} = {z xy(z = x, y χ)})
8 df-opab 4623 . 2 {x, y ψ} = {z xy(z = x, y ψ)}
9 df-opab 4623 . 2 {x, y χ} = {z xy(z = x, y χ)}
107, 8, 93eqtr4g 2410 1 (φ → {x, y ψ} = {x, y χ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  ⟨cop 4561  {copab 4622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-opab 4623 This theorem is referenced by:  opabbidv  4625  fnoprabg  5585  mpteq12f  5655
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