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Theorem r19.2zb 4324
Description: A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 4323. 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 4323 . . 3 ((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
21ex 405 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))
3 noel 4184 . . . . . . 7 ¬ 𝑥 ∈ ∅
43pm2.21i 117 . . . . . 6 (𝑥 ∈ ∅ → 𝜑)
54rgen 3099 . . . . 5 𝑥 ∈ ∅ 𝜑
6 raleq 3346 . . . . 5 (𝐴 = ∅ → (∀𝑥𝐴 𝜑 ↔ ∀𝑥 ∈ ∅ 𝜑))
75, 6mpbiri 250 . . . 4 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
87necon3bi 2994 . . 3 (¬ ∀𝑥𝐴 𝜑𝐴 ≠ ∅)
9 exsimpl 1831 . . . 4 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥 𝑥𝐴)
10 df-rex 3095 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
11 n0 4197 . . . 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 387   = wceq 1507  wex 1742  wcel 2050  wne 2968  wral 3089  wrex 3090  c0 4179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-11 2093  ax-12 2106  ax-ext 2751
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-dif 3833  df-nul 4180
This theorem is referenced by:  iinpreima  6662  utopbas  22547  clsk3nimkb  39750  radcnvrat  40059
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