Theorem 2rexbidva 3228

Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)

Hypothesis

Ref	Expression
2ralbidva.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
2rexbidva	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

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

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

Proof of Theorem 2rexbidva

Step	Hyp	Ref	Expression
1		2ralbidva.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))
2	1	anassrs 473	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 ↔ 𝜒))
3	2	rexbidva 3187	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
4	3	rexbidva 3187	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∃wrex 3089

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-rex 3090

This theorem is used by:  2reu4lem  4481  wrdl3s3  15127  bezoutlem2  16725  bezoutlem4  16727  vdwmc2  17169  lsmcom2  19881  lsmass  19895  lsmcomx  20082  lsmspsn  21371  hausdiag  23976  imasf1oxms  24820  mulsval  28507  mulscom  28537  addsdi  28553  mulsasslem3  28563  mulsunif2lem  28567  z12sge0  28881  istrkg2ld  28934  iscgra  29328  axeuclid  29553  elwwlks2  30570  elwspths2spth  30571  fusgr2wsp2nb  30947  shscom  31933  lsmssass  34006  sategoelfvb  36305  ltnmul  37092  nmulle  37093  3dim0  40654  islpln5  40732  islvol5  40776  isline2  40971  isline3  40973  paddcom  41010  cdlemg2cex  41788  prprspr2  48739  pgrpgt2nabl  49617  elbigolo1  49808