Theorem reximdv 2571

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)

Hypothesis

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

Assertion

Ref	Expression
reximdv	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reximdv

Step	Hyp	Ref	Expression
1		reximdv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	a1d 22	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3	2	reximdvai 2570	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2141  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  r19.12  2576  reusv3  4445  rexxfrd  4448  iunpw  4465  fvelima  5548  carden2bex  7166  prnmaddl  7452  prarloclem5  7462  prarloc2  7466  genprndl  7483  genprndu  7484  ltpopr  7557  recexprlemm  7586  recexprlemopl  7587  recexprlemopu  7589  recexprlem1ssl  7595  recexprlem1ssu  7596  cauappcvgprlemupu  7611  caucvgprlemupu  7634  caucvgprprlemupu  7662  caucvgsrlemoffres  7762  map2psrprg  7767  resqrexlemgt0  10984  subcn2  11274  bezoutlembz  11959  pythagtriplem19  12236  tgcl  12858  neiss  12944  ssnei2  12951  tgcnp  13003  cnptopco  13016  cnptopresti  13032  lmtopcnp  13044  blssexps  13223  blssex  13224  mopni3  13278  neibl  13285  metss  13288  metcnp3  13305  rescncf  13362  limcresi  13429