Theorem 2rexbidv 2432

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 2410	. 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
3	2	rexbidv 2410	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 104  ∃wrex 2389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-17 1487  ax-ial 1495

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-rex 2394

This theorem is referenced by:  f1oiso  5679  elrnmpog  5835  elrnmpo  5836  ralrnmpo  5837  rexrnmpo  5838  ovelrn  5871  eroveu  6472  genipv  7259  genpelxp  7261  genpelvl  7262  genpelvu  7263  axcnre  7610  apreap  8261  apreim  8277  bezoutlemnewy  11524  bezoutlema  11527  bezoutlemb  11528  txuni2  12261  txbas  12263  txdis1cn  12283