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Theorem prtlem100 33161
 Description: Lemma for prter3 33185. (Contributed by Rodolfo Medina, 19-Oct-2010.)
Assertion
Ref Expression
prtlem100 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))

Proof of Theorem prtlem100
StepHypRef Expression
1 anass 679 . . 3 (((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
2 eldifsn 4260 . . . 4 (𝑥 ∈ (𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
32anbi1i 727 . . 3 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)) ↔ ((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)))
4 ne0i 3880 . . . . . . 7 (𝐵𝑥𝑥 ≠ ∅)
54pm4.71ri 663 . . . . . 6 (𝐵𝑥 ↔ (𝑥 ≠ ∅ ∧ 𝐵𝑥))
65anbi1i 727 . . . . 5 ((𝐵𝑥𝜑) ↔ ((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑))
7 anass 679 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
86, 7bitri 263 . . . 4 ((𝐵𝑥𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
98anbi2i 726 . . 3 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
101, 3, 93bitr4ri 292 . 2 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)))
1110rexbii2 3021 1 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537  ∅c0 3874  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-v 3175  df-dif 3543  df-nul 3875  df-sn 4126 This theorem is referenced by: (None)
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