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Theorem ax12indalem 34732
Description: Lemma for ax12inda2 34734 and ax12inda 34735. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12indalem.1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Assertion
Ref Expression
ax12indalem (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))

Proof of Theorem ax12indalem
StepHypRef Expression
1 ax-1 6 . . . . . . . . 9 (∀𝑥𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜑))
21axc4i-o 34685 . . . . . . . 8 (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑))
32a1i 11 . . . . . . 7 (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)))
4 biidd 252 . . . . . . . 8 (∀𝑧 𝑧 = 𝑥 → (𝜑𝜑))
54dral1-o 34691 . . . . . . 7 (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑))
65imbi2d 329 . . . . . . . 8 (∀𝑧 𝑧 = 𝑥 → ((𝑥 = 𝑦 → ∀𝑧𝜑) ↔ (𝑥 = 𝑦 → ∀𝑥𝜑)))
76dral2-o 34717 . . . . . . 7 (∀𝑧 𝑧 = 𝑥 → (∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)))
83, 5, 73imtr4d 283 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
98aecoms-o 34689 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
109a1d 25 . . . 4 (∀𝑥 𝑥 = 𝑧 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑))))
1110a1d 25 . . 3 (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
1211adantr 472 . 2 ((∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
13 simplr 809 . . . . 5 ((((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ ¬ ∀𝑥 𝑥 = 𝑦) ∧ 𝑥 = 𝑦) → ¬ ∀𝑥 𝑥 = 𝑦)
14 aecom-o 34688 . . . . . . . . 9 (∀𝑧 𝑧 = 𝑥 → ∀𝑥 𝑥 = 𝑧)
1514con3i 150 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑧 𝑧 = 𝑥)
16 aecom-o 34688 . . . . . . . . 9 (∀𝑧 𝑧 = 𝑦 → ∀𝑦 𝑦 = 𝑧)
1716con3i 150 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → ¬ ∀𝑧 𝑧 = 𝑦)
18 axc9 2445 . . . . . . . . 9 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
1918imp 444 . . . . . . . 8 ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
2015, 17, 19syl2an 495 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
2120imp 444 . . . . . 6 (((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ 𝑥 = 𝑦) → ∀𝑧 𝑥 = 𝑦)
2221adantlr 753 . . . . 5 ((((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ ¬ ∀𝑥 𝑥 = 𝑦) ∧ 𝑥 = 𝑦) → ∀𝑧 𝑥 = 𝑦)
23 hbnae-o 34715 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
24 hba1-o 34684 . . . . . . 7 (∀𝑧 𝑥 = 𝑦 → ∀𝑧𝑧 𝑥 = 𝑦)
2523, 24hban 2273 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ∀𝑧(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦))
26 ax-c5 34670 . . . . . . 7 (∀𝑧 𝑥 = 𝑦𝑥 = 𝑦)
27 ax12indalem.1 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
2827imp 444 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2926, 28sylan2 492 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3025, 29alimdh 1892 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → (∀𝑧𝜑 → ∀𝑧𝑥(𝑥 = 𝑦𝜑)))
3113, 22, 30syl2anc 696 . . . 4 ((((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ ¬ ∀𝑥 𝑥 = 𝑦) ∧ 𝑥 = 𝑦) → (∀𝑧𝜑 → ∀𝑧𝑥(𝑥 = 𝑦𝜑)))
32 ax-11 2181 . . . . . 6 (∀𝑧𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑧(𝑥 = 𝑦𝜑))
33 hbnae-o 34715 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑧 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑧)
34 hbnae-o 34715 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑥 ¬ ∀𝑦 𝑦 = 𝑧)
3533, 34hban 2273 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑥(¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧))
36 hbnae-o 34715 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑧 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑧)
37 hbnae-o 34715 . . . . . . . . . 10 (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑧 ¬ ∀𝑦 𝑦 = 𝑧)
3836, 37hban 2273 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∀𝑧(¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧))
3938, 20nf5dh 2173 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → Ⅎ𝑧 𝑥 = 𝑦)
40 19.21t 2218 . . . . . . . 8 (Ⅎ𝑧 𝑥 = 𝑦 → (∀𝑧(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑧𝜑)))
4139, 40syl 17 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑧(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦 → ∀𝑧𝜑)))
4235, 41albidh 1940 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥𝑧(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
4332, 42syl5ib 234 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑧𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
4443ad2antrr 764 . . . 4 ((((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ ¬ ∀𝑥 𝑥 = 𝑦) ∧ 𝑥 = 𝑦) → (∀𝑧𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
4531, 44syld 47 . . 3 ((((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) ∧ ¬ ∀𝑥 𝑥 = 𝑦) ∧ 𝑥 = 𝑦) → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))
4645exp31 631 . 2 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
4712, 46pm2.61ian 866 1 (¬ ∀𝑦 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑧𝜑)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1628  wnf 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-c5 34670  ax-c4 34671  ax-c7 34672  ax-c10 34673  ax-c11 34674  ax-c9 34677
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857
This theorem is referenced by:  ax12inda2  34734
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