Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnopabeqd Structured version   Visualization version   GIF version

Theorem fnopabeqd 33821
Description: Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)
Hypothesis
Ref Expression
fnopabeqd.1 (𝜑𝐵 = 𝐶)
Assertion
Ref Expression
fnopabeqd (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem fnopabeqd
StepHypRef Expression
1 fnopabeqd.1 . . . 4 (𝜑𝐵 = 𝐶)
21eqeq2d 2816 . . 3 (𝜑 → (𝑦 = 𝐵𝑦 = 𝐶))
32anbi2d 616 . 2 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐴𝑦 = 𝐶)))
43opabbidv 4910 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  {copab 4906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-opab 4907
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator