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Theorem ax12v2-o 36084
 Description: Rederivation of ax-c15 36024 from ax12v 2174 (without using ax-c15 36024 or the full ax-12 2173). Thus, the hypothesis (ax12v 2174) provides an alternate axiom that can be used in place of ax-c15 36024. See also axc15 2440. (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 1968 . 2 𝑧 𝑧 = 𝑦
2 ax12v2-o.1 . . . . 5 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
3 equequ2 2029 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
43adantl 484 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑧𝑥 = 𝑦))
5 dveeq2-o 36068 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
65imp 409 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ∀𝑥 𝑧 = 𝑦)
7 nfa1-o 36050 . . . . . . . . 9 𝑥𝑥 𝑧 = 𝑦
83imbi1d 344 . . . . . . . . . 10 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
98sps-o 36043 . . . . . . . . 9 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
107, 9albid 2220 . . . . . . . 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 343 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
134, 12imbi12d 347 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
142, 13mpbii 235 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦𝑧 = 𝑦) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
1514ex 415 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
1615exlimdv 1930 . 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 208   ∧ wa 398  ∀wal 1531  ∃wex 1776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173  ax-13 2386  ax-c5 36018  ax-c4 36019  ax-c7 36020  ax-c10 36021  ax-c11 36022  ax-c9 36025 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781 This theorem is referenced by:  ax12a2-o  36085
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