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Theorem dvelimf-o 33015
Description: Proof of dvelimh 2323 that uses ax-c11 32973 but not ax-c15 32975, ax-c11n 32974, or ax-12 2033. Version of dvelimh 2323 using ax-c11 32973 instead of axc11 2301. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvelimf-o.1 (𝜑 → ∀𝑥𝜑)
dvelimf-o.2 (𝜓 → ∀𝑧𝜓)
dvelimf-o.3 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimf-o (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Proof of Theorem dvelimf-o
StepHypRef Expression
1 hba1-o 32983 . . . . 5 (∀𝑧(𝑧 = 𝑦𝜑) → ∀𝑧𝑧(𝑧 = 𝑦𝜑))
2 ax-c11 32973 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝑧(𝑧 = 𝑦𝜑) → ∀𝑥𝑧(𝑧 = 𝑦𝜑)))
32aecoms-o 32988 . . . . 5 (∀𝑥 𝑥 = 𝑧 → (∀𝑧𝑧(𝑧 = 𝑦𝜑) → ∀𝑥𝑧(𝑧 = 𝑦𝜑)))
41, 3syl5 33 . . . 4 (∀𝑥 𝑥 = 𝑧 → (∀𝑧(𝑧 = 𝑦𝜑) → ∀𝑥𝑧(𝑧 = 𝑦𝜑)))
54a1d 25 . . 3 (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑦𝜑) → ∀𝑥𝑧(𝑧 = 𝑦𝜑))))
6 hbnae-o 33014 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑧 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑧)
7 hbnae-o 33014 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
86, 7hban 2112 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑧(¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦))
9 hbnae-o 33014 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑧 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑧)
10 hbnae-o 33014 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
119, 10hban 2112 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥(¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦))
12 ax-c9 32976 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
1312imp 443 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
14 dvelimf-o.1 . . . . . . 7 (𝜑 → ∀𝑥𝜑)
1514a1i 11 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝜑 → ∀𝑥𝜑))
1611, 13, 15hbimd 2110 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ((𝑧 = 𝑦𝜑) → ∀𝑥(𝑧 = 𝑦𝜑)))
178, 16hbald 2027 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (∀𝑧(𝑧 = 𝑦𝜑) → ∀𝑥𝑧(𝑧 = 𝑦𝜑)))
1817ex 448 . . 3 (¬ ∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑦𝜑) → ∀𝑥𝑧(𝑧 = 𝑦𝜑))))
195, 18pm2.61i 174 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑦𝜑) → ∀𝑥𝑧(𝑧 = 𝑦𝜑)))
20 dvelimf-o.2 . . 3 (𝜓 → ∀𝑧𝜓)
21 dvelimf-o.3 . . 3 (𝑧 = 𝑦 → (𝜑𝜓))
2220, 21equsalh 2279 . 2 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
2322albii 1736 . 2 (∀𝑥𝑧(𝑧 = 𝑦𝜑) ↔ ∀𝑥𝜓)
2419, 22, 233imtr3g 282 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-c5 32969  ax-c4 32970  ax-c7 32971  ax-c10 32972  ax-c11 32973  ax-c9 32976
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by:  dveeq2-o  33019  dveeq1-o  33021  ax12el  33028
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