Theorem reximdv 2531

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 2530	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 1480  ∃wrex 2415

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419  df-rex 2420

This theorem is referenced by:  r19.12  2536  reusv3  4376  rexxfrd  4379  iunpw  4396  fvelima  5466  carden2bex  7038  prnmaddl  7291  prarloclem5  7301  prarloc2  7305  genprndl  7322  genprndu  7323  ltpopr  7396  recexprlemm  7425  recexprlemopl  7426  recexprlemopu  7428  recexprlem1ssl  7434  recexprlem1ssu  7435  cauappcvgprlemupu  7450  caucvgprlemupu  7473  caucvgprprlemupu  7501  caucvgsrlemoffres  7601  map2psrprg  7606  resqrexlemgt0  10785  subcn2  11073  bezoutlembz  11681  tgcl  12222  neiss  12308  ssnei2  12315  tgcnp  12367  cnptopco  12380  cnptopresti  12396  lmtopcnp  12408  blssexps  12587  blssex  12588  mopni3  12642  neibl  12649  metss  12652  metcnp3  12669  rescncf  12726  limcresi  12793