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Theorem prtlem100 34667
Description: Lemma for prter3 34690. (Contributed by Rodolfo Medina, 19-Oct-2010.)
Assertion
Ref Expression
prtlem100 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))

Proof of Theorem prtlem100
StepHypRef Expression
1 anass 454 . . 3 (((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
2 eldifsn 4454 . . . 4 (𝑥 ∈ (𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
32anbi1i 610 . . 3 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)) ↔ ((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)))
4 ne0i 4069 . . . . . . 7 (𝐵𝑥𝑥 ≠ ∅)
54pm4.71ri 550 . . . . . 6 (𝐵𝑥 ↔ (𝑥 ≠ ∅ ∧ 𝐵𝑥))
65anbi1i 610 . . . . 5 ((𝐵𝑥𝜑) ↔ ((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑))
7 anass 454 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
86, 7bitri 264 . . . 4 ((𝐵𝑥𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
98anbi2i 609 . . 3 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
101, 3, 93bitr4ri 293 . 2 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)))
1110rexbii2 3187 1 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wcel 2145  wne 2943  wrex 3062  cdif 3720  c0 4063  {csn 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rex 3067  df-v 3353  df-dif 3726  df-nul 4064  df-sn 4318
This theorem is referenced by: (None)
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