Theorem 2rexbidv 2522

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 2498	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
3	2	rexbidv 2498	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-rex 2481

This theorem is referenced by:  f1oiso  5873  elrnmpog  6035  elrnmpo  6036  ralrnmpo  6037  rexrnmpo  6038  ovelrn  6072  eroveu  6685  genipv  7576  genpelxp  7578  genpelvl  7579  genpelvu  7580  axcnre  7948  apreap  8614  apreim  8630  aprcl  8673  aptap  8677  bezoutlemnewy  12163  bezoutlema  12166  bezoutlemb  12167  pythagtriplem19  12451  pceu  12464  pcval  12465  pczpre  12466  pcdiv  12471  4sqlem2  12558  4sqlem3  12559  4sqlem4  12561  4sqexercise2  12568  4sqlemsdc  12569  4sq  12579  znunit  14215  txuni2  14492  txbas  14494  txdis1cn  14514  elply  14970  2sqlem2  15356  2sqlem8  15364  2sqlem9  15365