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Theorem prtlem100 34638
Description: Lemma for prter3 34661. (Contributed by Rodolfo Medina, 19-Oct-2010.)
Assertion
Ref Expression
prtlem100 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))

Proof of Theorem prtlem100
StepHypRef Expression
1 anass 456 . . 3 (((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
2 eldifsn 4508 . . . 4 (𝑥 ∈ (𝐴 ∖ {∅}) ↔ (𝑥𝐴𝑥 ≠ ∅))
32anbi1i 612 . . 3 ((𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)) ↔ ((𝑥𝐴𝑥 ≠ ∅) ∧ (𝐵𝑥𝜑)))
4 ne0i 4122 . . . . . . 7 (𝐵𝑥𝑥 ≠ ∅)
54pm4.71ri 552 . . . . . 6 (𝐵𝑥 ↔ (𝑥 ≠ ∅ ∧ 𝐵𝑥))
65anbi1i 612 . . . . 5 ((𝐵𝑥𝜑) ↔ ((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑))
7 anass 456 . . . . 5 (((𝑥 ≠ ∅ ∧ 𝐵𝑥) ∧ 𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
86, 7bitri 266 . . . 4 ((𝐵𝑥𝜑) ↔ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑)))
98anbi2i 611 . . 3 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥𝐴 ∧ (𝑥 ≠ ∅ ∧ (𝐵𝑥𝜑))))
101, 3, 93bitr4ri 295 . 2 ((𝑥𝐴 ∧ (𝐵𝑥𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ {∅}) ∧ (𝐵𝑥𝜑)))
1110rexbii2 3227 1 (∃𝑥𝐴 (𝐵𝑥𝜑) ↔ ∃𝑥 ∈ (𝐴 ∖ {∅})(𝐵𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wcel 2156  wne 2978  wrex 3097  cdif 3766  c0 4116  {csn 4370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-rex 3102  df-v 3393  df-dif 3772  df-nul 4117  df-sn 4371
This theorem is referenced by: (None)
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