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Theorem bnj1533 32185
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 32130 . . 3 (𝜃 → ∀𝑧(𝑧𝐵 → ¬ 𝑧𝐷))
3 bnj1533.3 . . . . . . . 8 𝐷 = {𝑧𝐴𝐶𝐸}
43rabeq2i 3473 . . . . . . 7 (𝑧𝐷 ↔ (𝑧𝐴𝐶𝐸))
54notbii 323 . . . . . 6 𝑧𝐷 ↔ ¬ (𝑧𝐴𝐶𝐸))
6 imnan 403 . . . . . 6 ((𝑧𝐴 → ¬ 𝐶𝐸) ↔ ¬ (𝑧𝐴𝐶𝐸))
7 nne 3018 . . . . . . 7 𝐶𝐸𝐶 = 𝐸)
87imbi2i 339 . . . . . 6 ((𝑧𝐴 → ¬ 𝐶𝐸) ↔ (𝑧𝐴𝐶 = 𝐸))
95, 6, 83bitr2i 302 . . . . 5 𝑧𝐷 ↔ (𝑧𝐴𝐶 = 𝐸))
109imbi2i 339 . . . 4 ((𝑧𝐵 → ¬ 𝑧𝐷) ↔ (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
11 bnj1533.2 . . . . . . 7 𝐵𝐴
1211sseli 3949 . . . . . 6 (𝑧𝐵𝑧𝐴)
1312imim1i 63 . . . . 5 ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸))
14 ax-1 6 . . . . . . . . 9 ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
1514anim1i 617 . . . . . . . 8 (((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵))
16 simpr 488 . . . . . . . . . 10 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝑧𝐵)
17 simpl 486 . . . . . . . . . 10 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
1816, 17mpd 15 . . . . . . . . 9 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → (𝑧𝐴𝐶 = 𝐸))
1918, 16jca 515 . . . . . . . 8 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → ((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵))
2015, 19impbii 212 . . . . . . 7 (((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵))
2120imbi1i 353 . . . . . 6 ((((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝐶 = 𝐸))
22 impexp 454 . . . . . 6 ((((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸)))
23 impexp 454 . . . . . 6 ((((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸)))
2421, 22, 233bitr3i 304 . . . . 5 (((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸)) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸)))
2513, 24mpbi 233 . . . 4 ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸))
2610, 25sylbi 220 . . 3 ((𝑧𝐵 → ¬ 𝑧𝐷) → (𝑧𝐵𝐶 = 𝐸))
272, 26sylg 1824 . 2 (𝜃 → ∀𝑧(𝑧𝐵𝐶 = 𝐸))
2827bnj1142 32122 1 (𝜃 → ∀𝑧𝐵 𝐶 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  {crab 3137  wss 3919
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-ral 3138  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936
This theorem is referenced by:  bnj1523  32404
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