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Theorem bnj1533 30176
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1533.1 (𝜃 → ∀𝑧𝐵 ¬ 𝑧𝐷)
bnj1533.2 𝐵𝐴
bnj1533.3 𝐷 = {𝑧𝐴𝐶𝐸}
Assertion
Ref Expression
bnj1533 (𝜃 → ∀𝑧𝐵 𝐶 = 𝐸)

Proof of Theorem bnj1533
StepHypRef Expression
1 bnj1533.1 . . . 4 (𝜃 → ∀𝑧𝐵 ¬ 𝑧𝐷)
21bnj1211 30122 . . 3 (𝜃 → ∀𝑧(𝑧𝐵 → ¬ 𝑧𝐷))
3 bnj1533.3 . . . . . . . . 9 𝐷 = {𝑧𝐴𝐶𝐸}
43rabeq2i 3170 . . . . . . . 8 (𝑧𝐷 ↔ (𝑧𝐴𝐶𝐸))
54notbii 309 . . . . . . 7 𝑧𝐷 ↔ ¬ (𝑧𝐴𝐶𝐸))
6 imnan 437 . . . . . . 7 ((𝑧𝐴 → ¬ 𝐶𝐸) ↔ ¬ (𝑧𝐴𝐶𝐸))
7 nne 2786 . . . . . . . 8 𝐶𝐸𝐶 = 𝐸)
87imbi2i 325 . . . . . . 7 ((𝑧𝐴 → ¬ 𝐶𝐸) ↔ (𝑧𝐴𝐶 = 𝐸))
95, 6, 83bitr2i 287 . . . . . 6 𝑧𝐷 ↔ (𝑧𝐴𝐶 = 𝐸))
109imbi2i 325 . . . . 5 ((𝑧𝐵 → ¬ 𝑧𝐷) ↔ (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
11 bnj1533.2 . . . . . . . 8 𝐵𝐴
1211sseli 3564 . . . . . . 7 (𝑧𝐵𝑧𝐴)
1312imim1i 61 . . . . . 6 ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸))
14 ax-1 6 . . . . . . . . . 10 ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
1514anim1i 590 . . . . . . . . 9 (((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵))
16 simpr 476 . . . . . . . . . . 11 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝑧𝐵)
17 simpl 472 . . . . . . . . . . 11 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
1816, 17mpd 15 . . . . . . . . . 10 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → (𝑧𝐴𝐶 = 𝐸))
1918, 16jca 553 . . . . . . . . 9 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → ((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵))
2015, 19impbii 198 . . . . . . . 8 (((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵))
2120imbi1i 338 . . . . . . 7 ((((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝐶 = 𝐸))
22 impexp 461 . . . . . . 7 ((((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸)))
23 impexp 461 . . . . . . 7 ((((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸)))
2421, 22, 233bitr3i 289 . . . . . 6 (((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸)) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸)))
2513, 24mpbi 219 . . . . 5 ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸))
2610, 25sylbi 206 . . . 4 ((𝑧𝐵 → ¬ 𝑧𝐷) → (𝑧𝐵𝐶 = 𝐸))
2726alimi 1730 . . 3 (∀𝑧(𝑧𝐵 → ¬ 𝑧𝐷) → ∀𝑧(𝑧𝐵𝐶 = 𝐸))
282, 27syl 17 . 2 (𝜃 → ∀𝑧(𝑧𝐵𝐶 = 𝐸))
2928bnj1142 30114 1 (𝜃 → ∀𝑧𝐵 𝐶 = 𝐸)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900   ⊆ wss 3540 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-ne 2782  df-ral 2901  df-rab 2905  df-in 3547  df-ss 3554 This theorem is referenced by:  bnj1523  30393
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