Theorem r19.2zb 4284

Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4283. 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 4283	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
2	1	ex 403	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))
3		noel 4149	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
4	3	pm2.21i 117	. . . . . 6 ⊢ (𝑥 ∈ ∅ → 𝜑)
5	4	rgen 3132	. . . . 5 ⊢ ∀𝑥 ∈ ∅ 𝜑
6		raleq 3351	. . . . 5 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑))
7	5, 6	mpbiri 250	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
8	7	necon3bi 3026	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
9		exsimpl 1971	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴)
10		df-rex 3124	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
11		n0 4161	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
12	9, 10, 11	3imtr4i 284	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
13	8, 12	ja 175	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) → 𝐴 ≠ ∅)
14	2, 13	impbii 201	1 ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658  ∃wex 1880   ∈ wcel 2166   ≠ wne 3000  ∀wral 3118  ∃wrex 3119  ∅c0 4145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-v 3417  df-dif 3802  df-nul 4146

This theorem is referenced by:  iinpreima  6595  utopbas  22410  clsk3nimkb  39179  radcnvrat  39354