Theorem rexrab 3654

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 3645	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	anbi1i 636	. . 3 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒))
4		anass 474	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜓) ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒)))
5	3, 4	bitri 278	. 2 ⊢ ((𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∧ 𝜒) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜒)))
6	5	rexbii2 3106	1 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜒 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  ∃wrex 3087  {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-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3088  df-rab 3414  df-v 3453

This theorem is used by:  wereu2  5648  frpomin  6336  wdom2d  9558  enfin2i  10380  infm3  12257  pmtrfrn  19652  pgpssslw  19808  ellspd  22088  1stcfb  23743  xkobval  23885  xkococn  23959  imasdsf1olem  24672  infdesc  27949  eqcuts2  28154  cutsun12  28158  cuteq0  28183  bdayons  28644  rusgrnumwwlks  30548  cvmliftlem15  36032  wsuclem  36557  poimirlem4  38510  poimirlem26  38532  poimirlem27  38533  rexrabdioph  43754  hbtlem6  44089  uhgrimisgrgric  48973  uspgrlimlem1  49030