Theorem nrexdv 2718

Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Hypothesis

Ref	Expression
nrexdv.1	⊢ ((φ ∧ x ∈ A) → ¬ ψ)

Assertion

Ref	Expression
nrexdv	⊢ (φ → ¬ ∃x ∈ A ψ)

Distinct variable group:   φ,x

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

Proof of Theorem nrexdv

Step	Hyp	Ref	Expression
1		nrexdv.1	. . 3 ⊢ ((φ ∧ x ∈ A) → ¬ ψ)
2	1	ralrimiva 2698	. 2 ⊢ (φ → ∀x ∈ A ¬ ψ)
3		ralnex 2625	. 2 ⊢ (∀x ∈ A ¬ ψ ↔ ¬ ∃x ∈ A ψ)
4	2, 3	sylib 188	1 ⊢ (φ → ¬ ∃x ∈ A ψ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  nndisjeq  4430  ltfinirr  4458  nnadjoin  4521  tfinnn  4535  vfinncvntsp  4550