Theorem exlimivv 1868

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)

Hypothesis

Ref	Expression
exlimivv.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
exlimivv	⊢ (∃𝑥∃𝑦𝜑 → 𝜓)

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem exlimivv

Step	Hyp	Ref	Expression
1		exlimivv.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	exlimiv 1577	. 2 ⊢ (∃𝑦𝜑 → 𝜓)
3	2	exlimiv 1577	1 ⊢ (∃𝑥∃𝑦𝜑 → 𝜓)

Syntax hints:   → wi 4  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-gen 1425  ax-ie2 1470  ax-17 1506

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  cgsex2g  2722  cgsex4g  2723  opabss  3992  copsexg  4166  elopab  4180  epelg  4212  0nelelxp  4568  elvvuni  4603  optocl  4615  xpsspw  4651  relopabi  4665  relop  4689  elreldm  4765  xpmlem  4959  dfco2a  5039  unielrel  5066  oprabid  5803  1stval2  6053  2ndval2  6054  xp1st  6063  xp2nd  6064  poxp  6129  rntpos  6154  dftpos4  6160  tpostpos  6161  tfrlem7  6214  th3qlem2  6532  ener  6673  domtr  6679  unen  6710  xpsnen  6715  mapen  6740  ltdcnq  7205  archnqq  7225  enq0tr  7242  nqnq0pi  7246  nqnq0  7249  nqpnq0nq  7261  nqnq0a  7262  nqnq0m  7263  nq0m0r  7264  nq0a0  7265  nq02m  7273  prarloc  7311  axaddcl  7672  axmulcl  7674  hashfacen  10579  bj-inex  13105