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Theorem rabbi 3112
Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 3179. (Contributed by NM, 25-Nov-2013.)
Assertion
Ref Expression
rabbi (∀𝑥𝐴 (𝜓𝜒) ↔ {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Proof of Theorem rabbi
StepHypRef Expression
1 abbi 2734 . 2 (∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)) ↔ {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐴𝜒)})
2 df-ral 2912 . . 3 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥(𝑥𝐴 → (𝜓𝜒)))
3 pm5.32 667 . . . 4 ((𝑥𝐴 → (𝜓𝜒)) ↔ ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
43albii 1744 . . 3 (∀𝑥(𝑥𝐴 → (𝜓𝜒)) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
52, 4bitri 264 . 2 (∀𝑥𝐴 (𝜓𝜒) ↔ ∀𝑥((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
6 df-rab 2916 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
7 df-rab 2916 . . 3 {𝑥𝐴𝜒} = {𝑥 ∣ (𝑥𝐴𝜒)}
86, 7eqeq12i 2635 . 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 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  {crab 2911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-rab 2916
This theorem is referenced by:  rabbidva  3179  kqfeq  21446  isr0  21459  bj-rabbida  32588  rabeq12f  33624  eq0rabdioph  36847  eqrabdioph  36848  lerabdioph  36876  eluzrabdioph  36877  ltrabdioph  36879  nerabdioph  36880  dvdsrabdioph  36881  undisjrab  38014  rabbida  38784  ioodvbdlimc1lem2  39475  ioodvbdlimc2lem  39477  fourierdlem89  39740  fourierdlem91  39742  fourierdlem100  39751  fourierdlem108  39759  fourierdlem112  39763  ovn0  40108  issmfdmpt  40285
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