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Theorem prtlem100 36195
 Description: Lemma for prter3 36218. (Contributed by Rodolfo Medina, 19-Oct-2010.)
Assertion
Ref Expression
prtlem100 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))

Proof of Theorem prtlem100
StepHypRef Expression
1 anass 472 . . 3 (((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
2 eldifsn 4680 . . . 4 (𝑥 ∈ (𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
32anbi1i 626 . . 3 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)) ↔ ((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)))
4 ne0i 4250 . . . . . . 7 (𝐵𝑥𝑥 ≠ ∅)
54pm4.71ri 564 . . . . . 6 (𝐵𝑥 ↔ (𝑥 ≠ ∅ ∧ 𝐵𝑥))
65anbi1i 626 . . . . 5 ((𝐵𝑥𝜑) ↔ ((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑))
7 anass 472 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
86, 7bitri 278 . . . 4 ((𝐵𝑥𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
98anbi2i 625 . . 3 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
101, 3, 93bitr4ri 307 . 2 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)))
1110rexbii2 3208 1 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107   ∖ cdif 3878  ∅c0 4243  {csn 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-rex 3112  df-v 3443  df-dif 3884  df-nul 4244  df-sn 4526 This theorem is referenced by: (None)
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