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Theorem prtlem100 38240
Description: Lemma for prter3 38263. (Contributed by Rodolfo Medina, 19-Oct-2010.)
Assertion
Ref Expression
prtlem100 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))

Proof of Theorem prtlem100
StepHypRef Expression
1 anass 468 . . 3 (((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
2 eldifsn 4785 . . . 4 (𝑥 ∈ (𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
32anbi1i 623 . . 3 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)) ↔ ((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)))
4 ne0i 4329 . . . . . . 7 (𝐵𝑥𝑥 ≠ ∅)
54pm4.71ri 560 . . . . . 6 (𝐵𝑥 ↔ (𝑥 ≠ ∅ ∧ 𝐵𝑥))
65anbi1i 623 . . . . 5 ((𝐵𝑥𝜑) ↔ ((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑))
7 anass 468 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
86, 7bitri 275 . . . 4 ((𝐵𝑥𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
98anbi2i 622 . . 3 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
101, 3, 93bitr4ri 304 . 2 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)))
1110rexbii2 3084 1 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2098  wne 2934  wrex 3064  cdif 3940  c0 4317  {csn 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rex 3065  df-v 3470  df-dif 3946  df-nul 4318  df-sn 4624
This theorem is referenced by: (None)
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