Theorem reximdv 2633

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 2632	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2202  ∃wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515  df-rex 2516

This theorem is referenced by:  r19.12  2639  reusv3  4557  rexxfrd  4560  iunpw  4577  fvelima  5697  carden2bex  7393  prnmaddl  7709  prarloclem5  7719  prarloc2  7723  genprndl  7740  genprndu  7741  ltpopr  7814  recexprlemm  7843  recexprlemopl  7844  recexprlemopu  7846  recexprlem1ssl  7852  recexprlem1ssu  7853  cauappcvgprlemupu  7868  caucvgprlemupu  7891  caucvgprprlemupu  7919  caucvgsrlemoffres  8019  map2psrprg  8024  resqrexlemgt0  11580  subcn2  11871  bezoutlembz  12574  pythagtriplem19  12854  mplsubgfileminv  14713  tgcl  14787  neiss  14873  ssnei2  14880  tgcnp  14932  cnptopco  14945  cnptopresti  14961  lmtopcnp  14973  blssexps  15152  blssex  15153  mopni3  15207  neibl  15214  metss  15217  metcnp3  15234  mpomulcn  15289  rescncf  15304  limcresi  15389  plyss  15461  umgrnloop0  15967  uhgr2edg  16056