Theorem exlimdvv 1949

Description: Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)

Hypothesis

Ref	Expression
exlimdvv.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
exlimdvv	⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → 𝜒))

Distinct variable groups:   𝜒,𝑥   𝜑,𝑥   𝜒,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem exlimdvv

Step	Hyp	Ref	Expression
1		exlimdvv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	exlimdv 1868	. 2 ⊢ (𝜑 → (∃𝑦𝜓 → 𝜒))
3	2	exlimdv 1868	1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-5 1496  ax-gen 1498  ax-ie2 1543  ax-17 1575

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  euotd  4376  opabssxpd  4791  funopg  5391  funopsn  5865  th3qlem1  6884  fundmen  7060  sbthlemi10  7249  addnq0mo  7778  mulnq0mo  7779  genprndl  7852  genprndu  7853  genpdisj  7854  mullocpr  7902  addsrmo  8074  mulsrmo  8075  cnm  8163  summodc  12098  fsum2dlemstep  12149  prodmodc  12293  fprod2dlemstep  12337  txbasval  15262  upgr1een  16249