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Theorem axc15 2445
Description: Derivation of set.mm's original ax-c15 36143 from ax-c11n 36142 and the shorter ax-12 2178 that has replaced it.

Theorem ax12 2446 shows the reverse derivation of ax-12 2178 from ax-c15 36143.

Normally, axc15 2445 should be used rather than ax-c15 36143, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 26-Mar-2023.) (New usage is discouraged.)

Assertion
Ref Expression
axc15 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Proof of Theorem axc15
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1972 . 2 𝑧 𝑧 = 𝑦
2 dveeq2 2397 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3 ax12v 2179 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
4 equeuclr 2030 . . . . . 6 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
54sps 2185 . . . . 5 (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
64imim1d 82 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑦𝜑)))
76al2imi 1817 . . . . . 6 (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
87imim2d 57 . . . . 5 (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
95, 8imim12d 81 . . . 4 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
102, 3, 9syl6mpi 67 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
1110exlimdv 1934 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
121, 11mpi 20 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  ax12  2446  ax12b  2447  equs5  2484  ax12vALT  2493  bj-ax12v3ALT  34094
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