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Theorem bnj1533 31457
 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 31403 . . 3 (𝜃 → ∀𝑧(𝑧𝐵 → ¬ 𝑧𝐷))
3 bnj1533.3 . . . . . . . 8 𝐷 = {𝑧𝐴𝐶𝐸}
43rabeq2i 3410 . . . . . . 7 (𝑧𝐷 ↔ (𝑧𝐴𝐶𝐸))
54notbii 312 . . . . . 6 𝑧𝐷 ↔ ¬ (𝑧𝐴𝐶𝐸))
6 imnan 390 . . . . . 6 ((𝑧𝐴 → ¬ 𝐶𝐸) ↔ ¬ (𝑧𝐴𝐶𝐸))
7 nne 3003 . . . . . . 7 𝐶𝐸𝐶 = 𝐸)
87imbi2i 328 . . . . . 6 ((𝑧𝐴 → ¬ 𝐶𝐸) ↔ (𝑧𝐴𝐶 = 𝐸))
95, 6, 83bitr2i 291 . . . . 5 𝑧𝐷 ↔ (𝑧𝐴𝐶 = 𝐸))
109imbi2i 328 . . . 4 ((𝑧𝐵 → ¬ 𝑧𝐷) ↔ (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
11 bnj1533.2 . . . . . . 7 𝐵𝐴
1211sseli 3823 . . . . . 6 (𝑧𝐵𝑧𝐴)
1312imim1i 63 . . . . 5 ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸))
14 ax-1 6 . . . . . . . . 9 ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
1514anim1i 608 . . . . . . . 8 (((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵))
16 simpr 479 . . . . . . . . . 10 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝑧𝐵)
17 simpl 476 . . . . . . . . . 10 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
1816, 17mpd 15 . . . . . . . . 9 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → (𝑧𝐴𝐶 = 𝐸))
1918, 16jca 507 . . . . . . . 8 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → ((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵))
2015, 19impbii 201 . . . . . . 7 (((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵))
2120imbi1i 341 . . . . . 6 ((((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝐶 = 𝐸))
22 impexp 443 . . . . . 6 ((((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸)))
23 impexp 443 . . . . . 6 ((((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸)))
2421, 22, 233bitr3i 293 . . . . 5 (((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸)) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸)))
2513, 24mpbi 222 . . . 4 ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸))
2610, 25sylbi 209 . . 3 ((𝑧𝐵 → ¬ 𝑧𝐷) → (𝑧𝐵𝐶 = 𝐸))
272, 26sylg 1921 . 2 (𝜃 → ∀𝑧(𝑧𝐵𝐶 = 𝐸))
2827bnj1142 31395 1 (𝜃 → ∀𝑧𝐵 𝐶 = 𝐸)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  {crab 3121   ⊆ wss 3798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-ne 3000  df-ral 3122  df-rab 3126  df-in 3805  df-ss 3812 This theorem is referenced by:  bnj1523  31674
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