Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opeq2OLD Structured version   Visualization version   GIF version

Theorem opeq2OLD 4769
 Description: Obsolete version of opeq2 4768 as of 25-May-2024. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
opeq2OLD (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Proof of Theorem opeq2OLD
StepHypRef Expression
1 eleq1 2880 . . . 4 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
21anbi2d 631 . . 3 (𝐴 = 𝐵 → ((𝐶 ∈ V ∧ 𝐴 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐵 ∈ V)))
3 preq2 4633 . . . 4 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
43preq2d 4639 . . 3 (𝐴 = 𝐵 → {{𝐶}, {𝐶, 𝐴}} = {{𝐶}, {𝐶, 𝐵}})
52, 4ifbieq1d 4451 . 2 (𝐴 = 𝐵 → if((𝐶 ∈ V ∧ 𝐴 ∈ V), {{𝐶}, {𝐶, 𝐴}}, ∅) = if((𝐶 ∈ V ∧ 𝐵 ∈ V), {{𝐶}, {𝐶, 𝐵}}, ∅))
6 dfopif 4763 . 2 𝐶, 𝐴⟩ = if((𝐶 ∈ V ∧ 𝐴 ∈ V), {{𝐶}, {𝐶, 𝐴}}, ∅)
7 dfopif 4763 . 2 𝐶, 𝐵⟩ = if((𝐶 ∈ V ∧ 𝐵 ∈ V), {{𝐶}, {𝐶, 𝐵}}, ∅)
85, 6, 73eqtr4g 2861 1 (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ∅c0 4246  ifcif 4428  {csn 4528  {cpr 4530  ⟨cop 4534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator