Theorem rexlimdv 2647

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Hypothesis

Ref	Expression
rexlimdv.1	⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

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

Proof of Theorem rexlimdv

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜒
3		rexlimdv.1	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
4	1, 2, 3	rexlimd 2645	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 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  ax-i5r 1581

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

This theorem is referenced by:  rexlimdva  2648  rexlimdv3a  2650  rexlimdva2  2651  rexlimdvw  2652  rexlimdvv  2655  ssorduni  4583  funcnvuni  5396  dffo3  5790  smoiun  6462  tfrlem9  6480  ordiso2  7225  axprecex  8090  recexap  8823  zdiv  9558  btwnz  9589  lbzbi  9840  imasmnd2  13525  imasgrp2  13687  imasrng  13959  imasring  14067  neibl  15205  metcnp3  15225  ushgredgedg  16065  ushgredgedgloop  16067