Theorem nfrexd 3371

Description: Deduction version of nfrex 3373. Usage of this theorem is discouraged because it depends on ax-13 2375. See nfrexdw 3308 for a version with a disjoint variable condition, but not requiring ax-13 2375. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfrald.1	⊢ Ⅎ𝑦𝜑
nfrald.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfrald.3	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfrexd	⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Proof of Theorem nfrexd

Step	Hyp	Ref	Expression
1		dfrex2 3071	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
2		nfrald.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfrald.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		nfrald.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	4	nfnd 1856	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
6	2, 3, 5	nfrald 3370	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓)
7	6	nfnd 1856	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
8	1, 7	nfxfrd 1851	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1780  Ⅎwnfc 2888  ∀wral 3059  ∃wrex 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069

This theorem is referenced by:  nfrex  3373  nfiundg  48906