Theorem 2rexbidva 3225

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 3184	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
4	3	rexbidva 3184	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 2145  ∃wrex 3086

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

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-rex 3087

This theorem is used by:  2reu4lem  4479  wrdl3s3  15036  bezoutlem2  16631  bezoutlem4  16633  vdwmc2  17072  lsmcom2  19783  lsmass  19797  lsmcomx  19984  lsmspsn  21269  hausdiag  23872  imasf1oxms  24716  mulsval  28375  mulscom  28405  addsdi  28421  mulsasslem3  28431  mulsunif2lem  28435  z12sge0  28749  istrkg2ld  28802  iscgra  29196  axeuclid  29421  elwwlks2  30438  elwspths2spth  30439  fusgr2wsp2nb  30815  shscom  31801  lsmssass  33832  sategoelfvb  35999  ltnmul  36797  nmulle  36798  3dim0  40331  islpln5  40409  islvol5  40453  isline2  40648  isline3  40650  paddcom  40687  cdlemg2cex  41465  prprspr2  48419  pgrpgt2nabl  49297  elbigolo1  49488