Theorem rexrab 2893

Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab.1	⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))

Assertion

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

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

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

Proof of Theorem rexrab

Step	Hyp	Ref	Expression
1		ralab.1	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
2	1	elrab 2886	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	anbi1i 455	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒))
4		anass 399	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒)))
5	3, 4	bitri 183	. 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒)))
6	5	rexbii2 2481	1 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2141  ∃wrex 2449  {crab 2452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732

This theorem is referenced by: (None)