Theorem nrexdv 2412

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 2392	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝜓)
3		ralnex 2316	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓)
4	2, 3	sylib 127	1 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-17 1419

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  ltpopr  6693  cauappcvgprlemladdru  6754  cauappcvgprlemladdrl  6755  caucvgprlemladdrl  6776  caucvgprprlemaddq  6806