Theorem rexrab2 3696

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexrab2	⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒))

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜒,𝑥   𝜑,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)

Proof of Theorem rexrab2

Step	Hyp	Ref	Expression
1		df-rab 3150	. . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
2	1	rexeqi 3417	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓)
3		ralab2.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
4	3	rexab2 3694	. 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓 ↔ ∃𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒))
5		anass 471	. . . 4 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ (𝑦 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
6	5	exbii 1847	. . 3 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
7		df-rex 3147	. . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒)))
8	6, 7	bitr4i 280	. 2 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒))
9	2, 4, 8	3bitri 299	1 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∃wex 1779   ∈ wcel 2113  {cab 2802  ∃wrex 3142  {crab 3145

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-rex 3147  df-rab 3150

This theorem is referenced by:  frminex  5538  sstotbnd3  35058