Theorem r19.2zb 4423

Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4422. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)

Assertion

Ref	Expression
r19.2zb	⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.2zb

Step	Hyp	Ref	Expression
1		r19.2z 4422	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
2	1	ex 412	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))
3		noel 4261	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
4	3	pm2.21i 119	. . . . . 6 ⊢ (𝑥 ∈ ∅ → 𝜑)
5	4	rgen 3073	. . . . 5 ⊢ ∀𝑥 ∈ ∅ 𝜑
6		raleq 3333	. . . . 5 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑))
7	5, 6	mpbiri 257	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
8	7	necon3bi 2969	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
9		exsimpl 1872	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴)
10		df-rex 3069	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
11		n0 4277	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
12	9, 10, 11	3imtr4i 291	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
13	8, 12	ja 186	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) → 𝐴 ≠ ∅)
14	2, 13	impbii 208	1 ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-dif 3886  df-nul 4254

This theorem is referenced by:  iinpreima  6928  utopbas  23295  clsk3nimkb  41539  radcnvrat  41821