Theorem rabbi 3442

Description: Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbii 3418. (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 2830	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
2		df-rab 3414	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
3		df-rab 3414	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)}
4	2, 3	eqeq12i 2779	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜒)})
5		df-ral 3078	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
6		pm5.32 584	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
7	6	albii 1852	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
8	5, 7	bitri 278	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
9	1, 4, 8	3bitr4ri 307	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  {crab 3413

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-ral 3078  df-rab 3414

This theorem is used by:  kqfeq  24036  isr0  24049  rabeq12f  39069  eq0rabdioph  43766  eqrabdioph  43767  lerabdioph  43791  eluzrabdioph  43792  ltrabdioph  43794  nerabdioph  43795  dvdsrabdioph  43796  undisjrab  45275  ioodvbdlimc1lem2  46911  ioodvbdlimc2lem  46913  fourierdlem89  47174  fourierdlem91  47176  fourierdlem100  47185  fourierdlem108  47193  fourierdlem112  47197  ovn0  47545  issmfdmpt  47727  line2x  49835  line2y  49836