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Theorem r19.2zb 4206
Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4205. 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 4205 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
21ex 449 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
3 noel 4063 . . . . . . 7 ¬ 𝑥 ∈ ∅
43pm2.21i 116 . . . . . 6 (𝑥 ∈ ∅ → 𝜑)
54rgen 3061 . . . . 5 𝑥 ∈ ∅ 𝜑
6 raleq 3278 . . . . 5 (𝐴 = ∅ → (∀𝑥𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑))
75, 6mpbiri 248 . . . 4 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
87necon3bi 2959 . . 3 (¬ ∀𝑥𝐴 𝜑𝐴 ≠ ∅)
9 exsimpl 1944 . . . 4 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥 𝑥𝐴)
10 df-rex 3057 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
11 n0 4075 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
129, 10, 113imtr4i 281 . . 3 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
138, 12ja 173 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑) → 𝐴 ≠ ∅)
142, 13impbii 199 1 (𝐴 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2140  wne 2933  wral 3051  wrex 3052  c0 4059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-v 3343  df-dif 3719  df-nul 4060
This theorem is referenced by:  iinpreima  6510  utopbas  22261  clsk3nimkb  38859  radcnvrat  39034
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