Theorem rexlimdv 2582

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 1516	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1516	. 2 ⊢ Ⅎ𝑥𝜒
3		rexlimdv.1	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
4	1, 2, 3	rexlimd 2580	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2136  ∃wrex 2445

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  rexlimdva  2583  rexlimdv3a  2585  rexlimdva2  2586  rexlimdvw  2587  rexlimdvv  2590  ssorduni  4464  funcnvuni  5257  dffo3  5632  smoiun  6269  tfrlem9  6287  ordiso2  7000  axprecex  7821  recexap  8550  zdiv  9279  btwnz  9310  lbzbi  9554  neibl  13141  metcnp3  13161