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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 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.2zb
StepHypRef Expression
1 r19.2z 4283 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
21ex 403 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
3 noel 4149 . . . . . . 7 ¬ 𝑥 ∈ ∅
43pm2.21i 117 . . . . . 6 (𝑥 ∈ ∅ → 𝜑)
54rgen 3132 . . . . 5 𝑥 ∈ ∅ 𝜑
6 raleq 3351 . . . . 5 (𝐴 = ∅ → (∀𝑥𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑))
75, 6mpbiri 250 . . . 4 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
87necon3bi 3026 . . 3 (¬ ∀𝑥𝐴 𝜑𝐴 ≠ ∅)
9 exsimpl 1971 . . . 4 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥 𝑥𝐴)
10 df-rex 3124 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
11 n0 4161 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
129, 10, 113imtr4i 284 . . 3 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
138, 12ja 175 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑) → 𝐴 ≠ ∅)
142, 13impbii 201 1 (𝐴 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
Colors of variables: wff setvar class
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
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