Theorem 2rexbidv 2555

Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)

Hypothesis

Ref	Expression
2ralbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

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

Proof of Theorem 2rexbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	rexbidv 2531	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
3	2	rexbidv 2531	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-rex 2514

This theorem is referenced by:  f1oiso  5949  elrnmpog  6116  elrnmpo  6117  ralrnmpo  6118  rexrnmpo  6119  ovelrn  6153  eroveu  6771  genipv  7692  genpelxp  7694  genpelvl  7695  genpelvu  7696  axcnre  8064  apreap  8730  apreim  8746  aprcl  8789  aptap  8793  bezoutlemnewy  12512  bezoutlema  12515  bezoutlemb  12516  pythagtriplem19  12800  pceu  12813  pcval  12814  pczpre  12815  pcdiv  12820  4sqlem2  12907  4sqlem3  12908  4sqlem4  12910  4sqexercise2  12917  4sqlemsdc  12918  4sq  12928  znunit  14617  txuni2  14924  txbas  14926  txdis1cn  14946  elply  15402  2sqlem2  15788  2sqlem8  15796  2sqlem9  15797  upgredg  15936