Theorem r19.2zb 4466

Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4465. 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 4465	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
2	1	ex 417	. 2 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))
3		rzal 4460	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
4	3	necon3bi 2990	. . 3 ⊢ (¬ ∀𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
5		rexn0 4462	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
6	4, 5	ja 188	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑) → 𝐴 ≠ ∅)
7	2, 6	impbii 212	1 ⊢ (𝐴 ≠ ∅ ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-ne 2965  df-ral 3086  df-rex 3096  df-dif 3916  df-nul 4295

This theorem is referenced by:  iinpreima  7065  utopbas  24361  clsk3nimkb  44692  radcnvrat  44950