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Theorem r19.2zb 4452
Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4451. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
Assertion
Ref Expression
r19.2zb (𝐴 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.2zb
StepHypRef Expression
1 r19.2z 4451 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
21ex 413 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
3 noel 4289 . . . . . . 7 ¬ 𝑥 ∈ ∅
43pm2.21i 119 . . . . . 6 (𝑥 ∈ ∅ → 𝜑)
54rgen 3065 . . . . 5 𝑥 ∈ ∅ 𝜑
6 raleq 3308 . . . . 5 (𝐴 = ∅ → (∀𝑥𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑))
75, 6mpbiri 257 . . . 4 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
87necon3bi 2969 . . 3 (¬ ∀𝑥𝐴 𝜑𝐴 ≠ ∅)
9 exsimpl 1871 . . . 4 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥 𝑥𝐴)
10 df-rex 3073 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
11 n0 4305 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
129, 10, 113imtr4i 291 . . 3 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
138, 12ja 186 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑) → 𝐴 ≠ ∅)
142, 13impbii 208 1 (𝐴 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wne 2942  wral 3063  wrex 3072  c0 4281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-ral 3064  df-rex 3073  df-dif 3912  df-nul 4282
This theorem is referenced by:  iinpreima  7017  utopbas  23583  clsk3nimkb  42292  radcnvrat  42574
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