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Theorem ax12v2-o 34553
Description: Rederivation of ax-c15 34493 from ax12v 2088 (without using ax-c15 34493 or the full ax-12 2087). Thus, the hypothesis (ax12v 2088) provides an alternate axiom that can be used in place of ax-c15 34493. See also axc15 2339. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12v2-o.1 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
Assertion
Ref Expression
ax12v2-o (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12v2-o
StepHypRef Expression
1 ax6ev 1947 . 2 𝑧 𝑧 = 𝑦
2 ax12v2-o.1 . . . . 5 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
3 equequ2 1999 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
43adantl 481 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑧𝑥 = 𝑦))
5 dveeq2-o 34537 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
65imp 444 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ∀𝑥 𝑧 = 𝑦)
7 nfa1-o 34519 . . . . . . . . 9 𝑥𝑥 𝑧 = 𝑦
83imbi1d 330 . . . . . . . . . 10 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
98sps-o 34512 . . . . . . . . 9 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
107, 9albid 2128 . . . . . . . 8 (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
116, 10syl 17 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
1211imbi2d 329 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
134, 12imbi12d 333 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
142, 13mpbii 223 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
1514ex 449 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
1615exlimdv 1901 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
171, 16mpi 20 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-c5 34487  ax-c4 34488  ax-c7 34489  ax-c10 34490  ax-c11 34491  ax-c9 34494
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by:  ax12a2-o  34554
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