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

Theorem rabbi 3251
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 3320. (Contributed by NM, 25-Nov-2013.)
Assertion
Ref Expression
rabbi (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbi
StepHypRef Expression
1 abbi 2867 . 2 (∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)) ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
2 df-ral 3047 . . 3 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥(𝑥𝐴 → (𝜓𝜒)))
3 pm5.32 671 . . . 4 ((𝑥𝐴 → (𝜓𝜒)) ↔ ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43albii 1888 . . 3 (∀𝑥(𝑥𝐴 → (𝜓𝜒)) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
52, 4bitri 264 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
6 df-rab 3051 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
7 df-rab 3051 . . 3 {𝑥𝐴𝜒} = {𝑥 ∣ (𝑥𝐴𝜒)}
86, 7eqeq12i 2766 . 2 ({𝑥𝐴𝜓} = {𝑥𝐴𝜒} ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
91, 5, 83bitr4i 292 1 (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1622   = wceq 1624  wcel 2131  {cab 2738  wral 3042  {crab 3046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-ral 3047  df-rab 3051
This theorem is referenced by:  rabbidva  3320  kqfeq  21721  isr0  21734  bj-rabbida  33212  rabeq12f  34270  eq0rabdioph  37834  eqrabdioph  37835  lerabdioph  37863  eluzrabdioph  37864  ltrabdioph  37866  nerabdioph  37867  dvdsrabdioph  37868  undisjrab  38999  rabbida  39765  ioodvbdlimc1lem2  40642  ioodvbdlimc2lem  40644  fourierdlem89  40907  fourierdlem91  40909  fourierdlem100  40918  fourierdlem108  40926  fourierdlem112  40930  ovn0  41278  issmfdmpt  41455
  Copyright terms: Public domain W3C validator