Theorem rabbi 3309

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 3402. (Contributed by NM, 25-Nov-2013.)

Assertion

Ref	Expression
rabbi	⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Proof of Theorem rabbi

Step	Hyp	Ref	Expression
1		abbi 2811	. 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)})
2		df-ral 3068	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
3		pm5.32 573	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	3	albii 1823	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
5	2, 4	bitri 274	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
6		df-rab 3072	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
7		df-rab 3072	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)}
8	6, 7	eqeq12i 2756	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)})
9	1, 5, 8	3bitr4i 302	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  {crab 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-ral 3068  df-rab 3072

This theorem is referenced by:  rabbida  3398  rabbidvaOLD  3403  kqfeq  22783  isr0  22796  rabeq12f  36242  eq0rabdioph  40514  eqrabdioph  40515  lerabdioph  40543  eluzrabdioph  40544  ltrabdioph  40546  nerabdioph  40547  dvdsrabdioph  40548  undisjrab  41813  ioodvbdlimc1lem2  43363  ioodvbdlimc2lem  43365  fourierdlem89  43626  fourierdlem91  43628  fourierdlem100  43637  fourierdlem108  43645  fourierdlem112  43649  ovn0  43994  issmfdmpt  44171  line2x  45988  line2y  45989