Theorem 2rexbidva 3220

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 467	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 ↔ 𝜒))
3	2	rexbidva 3177	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
4	3	rexbidva 3177	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∃wrex 3070

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-rex 3071

This theorem is referenced by:  2reu4lem  4522  wrdl3s3  15001  bezoutlem2  16577  bezoutlem4  16579  vdwmc2  17017  lsmcom2  19673  lsmass  19687  lsmcomx  19874  lsmspsn  21083  hausdiag  23653  imasf1oxms  24502  mulsval  28135  mulscom  28165  addsdi  28181  mulsasslem3  28191  mulsunif2lem  28195  istrkg2ld  28468  iscgra  28817  axeuclid  28978  elwwlks2  29986  elwspths2spth  29987  fusgr2wsp2nb  30353  shscom  31338  lsmssass  33430  sategoelfvb  35424  3dim0  39459  islpln5  39537  islvol5  39581  isline2  39776  isline3  39778  paddcom  39815  cdlemg2cex  40593  prprspr2  47505  pgrpgt2nabl  48282  elbigolo1  48478