Theorem 2rexbidva 3299

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

Hypothesis

Ref	Expression
2rexbidva.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		2rexbidva.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))
2	1	anassrs 470	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 ↔ 𝜒))
3	2	rexbidva 3296	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
4	3	rexbidva 3296	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-rex 3144

This theorem is referenced by:  2reu4lem  4464  wrdl3s3  14320  bezoutlem2  15882  bezoutlem4  15884  vdwmc2  16309  lsmcom2  18774  lsmass  18789  lsmcomx  18970  lsmspsn  19850  hausdiag  22247  imasf1oxms  23093  istrkg2ld  26240  iscgra  26589  axeuclid  26743  elwwlks2  27739  elwspths2spth  27740  fusgr2wsp2nb  28107  shscom  29090  sategoelfvb  32661  3dim0  36587  islpln5  36665  islvol5  36709  isline2  36904  isline3  36906  paddcom  36943  cdlemg2cex  37721  prprspr2  43674  pgrpgt2nabl  44408  elbigolo1  44611