Theorem reximdv 2634

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 2633	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2202  ∃wrex 2512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516  df-rex 2517

This theorem is referenced by:  r19.12  2640  reusv3  4563  rexxfrd  4566  iunpw  4583  fvelima  5706  carden2bex  7437  prnmaddl  7753  prarloclem5  7763  prarloc2  7767  genprndl  7784  genprndu  7785  ltpopr  7858  recexprlemm  7887  recexprlemopl  7888  recexprlemopu  7890  recexprlem1ssl  7896  recexprlem1ssu  7897  cauappcvgprlemupu  7912  caucvgprlemupu  7935  caucvgprprlemupu  7963  caucvgsrlemoffres  8063  map2psrprg  8068  resqrexlemgt0  11643  subcn2  11934  bezoutlembz  12638  pythagtriplem19  12918  mplsubgfileminv  14784  tgcl  14858  neiss  14944  ssnei2  14951  tgcnp  15003  cnptopco  15016  cnptopresti  15032  lmtopcnp  15044  blssexps  15223  blssex  15224  mopni3  15278  neibl  15285  metss  15288  metcnp3  15305  mpomulcn  15360  rescncf  15375  limcresi  15460  plyss  15532  umgrnloop0  16041  uhgr2edg  16130