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

Proof of Theorem rabbi

Step	Hyp	Ref	Expression
1		abbi 2867	. 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)) ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)})
2		df-ral 3047	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
3		pm5.32 671	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	3	albii 1888	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
5	2, 4	bitri 264	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
6		df-rab 3051	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
7		df-rab 3051	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)}
8	6, 7	eqeq12i 2766	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)})
9	1, 5, 8	3bitr4i 292	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

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