Theorem reximdv 2631

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 2630	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2200  ∃wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  r19.12  2637  reusv3  4551  rexxfrd  4554  iunpw  4571  fvelima  5687  carden2bex  7373  prnmaddl  7688  prarloclem5  7698  prarloc2  7702  genprndl  7719  genprndu  7720  ltpopr  7793  recexprlemm  7822  recexprlemopl  7823  recexprlemopu  7825  recexprlem1ssl  7831  recexprlem1ssu  7832  cauappcvgprlemupu  7847  caucvgprlemupu  7870  caucvgprprlemupu  7898  caucvgsrlemoffres  7998  map2psrprg  8003  resqrexlemgt0  11546  subcn2  11837  bezoutlembz  12540  pythagtriplem19  12820  mplsubgfileminv  14679  tgcl  14753  neiss  14839  ssnei2  14846  tgcnp  14898  cnptopco  14911  cnptopresti  14927  lmtopcnp  14939  blssexps  15118  blssex  15119  mopni3  15173  neibl  15180  metss  15183  metcnp3  15200  mpomulcn  15255  rescncf  15270  limcresi  15355  plyss  15427  umgrnloop0  15932  uhgr2edg  16019