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Theorem bnj1533 30657
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 30603 . . 3 (𝜃 → ∀𝑧(𝑧𝐵 → ¬ 𝑧𝐷))
3 bnj1533.3 . . . . . . . 8 𝐷 = {𝑧𝐴𝐶𝐸}
43rabeq2i 3186 . . . . . . 7 (𝑧𝐷 ↔ (𝑧𝐴𝐶𝐸))
54notbii 310 . . . . . 6 𝑧𝐷 ↔ ¬ (𝑧𝐴𝐶𝐸))
6 imnan 438 . . . . . 6 ((𝑧𝐴 → ¬ 𝐶𝐸) ↔ ¬ (𝑧𝐴𝐶𝐸))
7 nne 2794 . . . . . . 7 𝐶𝐸𝐶 = 𝐸)
87imbi2i 326 . . . . . 6 ((𝑧𝐴 → ¬ 𝐶𝐸) ↔ (𝑧𝐴𝐶 = 𝐸))
95, 6, 83bitr2i 288 . . . . 5 𝑧𝐷 ↔ (𝑧𝐴𝐶 = 𝐸))
109imbi2i 326 . . . 4 ((𝑧𝐵 → ¬ 𝑧𝐷) ↔ (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
11 bnj1533.2 . . . . . . 7 𝐵𝐴
1211sseli 3583 . . . . . 6 (𝑧𝐵𝑧𝐴)
1312imim1i 63 . . . . 5 ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸))
14 ax-1 6 . . . . . . . . 9 ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
1514anim1i 591 . . . . . . . 8 (((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵))
16 simpr 477 . . . . . . . . . 10 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝑧𝐵)
17 simpl 473 . . . . . . . . . 10 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → (𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)))
1816, 17mpd 15 . . . . . . . . 9 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → (𝑧𝐴𝐶 = 𝐸))
1918, 16jca 554 . . . . . . . 8 (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → ((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵))
2015, 19impbii 199 . . . . . . 7 (((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵))
2120imbi1i 339 . . . . . 6 ((((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ (((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝐶 = 𝐸))
22 impexp 462 . . . . . 6 ((((𝑧𝐴𝐶 = 𝐸) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ ((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸)))
23 impexp 462 . . . . . 6 ((((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) ∧ 𝑧𝐵) → 𝐶 = 𝐸) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸)))
2421, 22, 233bitr3i 290 . . . . 5 (((𝑧𝐴𝐶 = 𝐸) → (𝑧𝐵𝐶 = 𝐸)) ↔ ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸)))
2513, 24mpbi 220 . . . 4 ((𝑧𝐵 → (𝑧𝐴𝐶 = 𝐸)) → (𝑧𝐵𝐶 = 𝐸))
2610, 25sylbi 207 . . 3 ((𝑧𝐵 → ¬ 𝑧𝐷) → (𝑧𝐵𝐶 = 𝐸))
272, 26sylg 1747 . 2 (𝜃 → ∀𝑧(𝑧𝐵𝐶 = 𝐸))
2827bnj1142 30595 1 (𝜃 → ∀𝑧𝐵 𝐶 = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  {crab 2911  wss 3559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2791  df-ral 2912  df-rab 2916  df-in 3566  df-ss 3573
This theorem is referenced by:  bnj1523  30874
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