Theorem rabbi 3425

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii 3400. (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		abbib 2798	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
2		df-rab 3395	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
3		df-rab 3395	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)}
4	2, 3	eqeq12i 2747	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)})
5		df-ral 3045	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
6		pm5.32 573	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
7	6	albii 1819	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
8	5, 7	bitri 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
9	1, 4, 8	3bitr4ri 304	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  {crab 3394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-ral 3045  df-rab 3395

This theorem is referenced by:  rabbidaOLD  3433  kqfeq  23609  isr0  23622  rabeq12f  38137  eq0rabdioph  42749  eqrabdioph  42750  lerabdioph  42778  eluzrabdioph  42779  ltrabdioph  42781  nerabdioph  42782  dvdsrabdioph  42783  undisjrab  44279  ioodvbdlimc1lem2  45913  ioodvbdlimc2lem  45915  fourierdlem89  46176  fourierdlem91  46178  fourierdlem100  46187  fourierdlem108  46195  fourierdlem112  46199  ovn0  46547  issmfdmpt  46729  line2x  48739  line2y  48740