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Theorem oprabbii 5565
 Description: Equivalent wff's yield equal operation class abstractions. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors, 28-May-1995.) (Revised by set.mm contributors, 24-Jul-2012.)
Hypothesis
Ref Expression
oprabbii.1 (φψ)
Assertion
Ref Expression
oprabbii {x, y, z φ} = {x, y, z ψ}
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem oprabbii
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 eqid 2353 . 2 w = w
2 oprabbii.1 . . . 4 (φψ)
32a1i 10 . . 3 (w = w → (φψ))
43oprabbidv 5564 . 2 (w = w → {x, y, z φ} = {x, y, z ψ})
51, 4ax-mp 8 1 {x, y, z φ} = {x, y, z ψ}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642  {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:  oprab4  5566  oprabbi2i  5647  mpt2v  5719
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