Theorem nrexdv 1271

Description: Deduction adding restricted existential quantifier to negated wff.

Hypothesis

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

Assertion

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

Distinct variable group(s):   φ,x

Proof of Theorem nrexdv

Step	Hyp	Ref	Expression
1		nrexdv.1	. . . 4 ⊢ ((φ ∧ x ∈ A) → ¬ ψ)
2	1	exp 291	. . 3 ⊢ (φ → (x ∈ A → ¬ ψ))
3	2	r19.21aiv 1259	. 2 ⊢ (φ → ∀x ∈ A ¬ ψ)
4		ralnex 1209	. 2 ⊢ (∀x ∈ A ¬ ψ ↔ ¬ ∃x ∈ A ψ)
5	3, 4	sylib 173	1 ⊢ (φ → ¬ ∃x ∈ A ψ)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  class2set 1747  peano5 2394  oalimcl 3162  setind 3492  cardlim 3657  cardaleph 3690

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

metamath.org