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Theorem r19.2zb 4437
Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4436. 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 4436 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
21ex 413 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
3 noel 4293 . . . . . . 7 ¬ 𝑥 ∈ ∅
43pm2.21i 119 . . . . . 6 (𝑥 ∈ ∅ → 𝜑)
54rgen 3145 . . . . 5 𝑥 ∈ ∅ 𝜑
6 raleq 3403 . . . . 5 (𝐴 = ∅ → (∀𝑥𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑))
75, 6mpbiri 259 . . . 4 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
87necon3bi 3039 . . 3 (¬ ∀𝑥𝐴 𝜑𝐴 ≠ ∅)
9 exsimpl 1860 . . . 4 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥 𝑥𝐴)
10 df-rex 3141 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
11 n0 4307 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
129, 10, 113imtr4i 293 . . 3 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
138, 12ja 187 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑) → 𝐴 ≠ ∅)
142, 13impbii 210 1 (𝐴 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-dif 3936  df-nul 4289
This theorem is referenced by:  iinpreima  6829  utopbas  22771  clsk3nimkb  40268  radcnvrat  40523
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