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Theorem oprabbid 5563
 Description: Equivalent wff's yield equal operation class abstractions (deduction rule). (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
oprabbid.1 xφ
oprabbid.2 yφ
oprabbid.3 zφ
oprabbid.4 (φ → (ψχ))
Assertion
Ref Expression
oprabbid (φ → {x, y, z ψ} = {x, y, z χ})
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)   χ(x,y,z)

Proof of Theorem oprabbid
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 oprabbid.1 . . . 4 xφ
2 oprabbid.2 . . . . 5 yφ
3 oprabbid.3 . . . . . 6 zφ
4 oprabbid.4 . . . . . . 7 (φ → (ψχ))
54anbi2d 684 . . . . . 6 (φ → ((w = x, y, z ψ) ↔ (w = x, y, z χ)))
63, 5exbid 1773 . . . . 5 (φ → (z(w = x, y, z ψ) ↔ z(w = x, y, z χ)))
72, 6exbid 1773 . . . 4 (φ → (yz(w = x, y, z ψ) ↔ yz(w = x, y, z χ)))
81, 7exbid 1773 . . 3 (φ → (xyz(w = x, y, z ψ) ↔ xyz(w = x, y, z χ)))
98abbidv 2467 . 2 (φ → {w xyz(w = x, y, z ψ)} = {w xyz(w = x, y, z χ)})
10 df-oprab 5528 . 2 {x, y, z ψ} = {w xyz(w = x, y, z ψ)}
11 df-oprab 5528 . 2 {x, y, z χ} = {w xyz(w = x, y, z χ)}
129, 10, 113eqtr4g 2410 1 (φ → {x, y, z ψ} = {x, y, z 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  {coprab 5527 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-oprab 5528 This theorem is referenced by:  oprabbidv  5564  mpt2eq123  5661
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