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Theorem axc15 2422
Description: Derivation of set.mm's original ax-c15 36903 from ax-c11n 36902 and the shorter ax-12 2171 that has replaced it.

Theorem ax12 2423 shows the reverse derivation of ax-12 2171 from ax-c15 36903.

Normally, axc15 2422 should be used rather than ax-c15 36903, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2372. (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 1973 . 2 𝑧 𝑧 = 𝑦
2 dveeq2 2378 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3 ax12v 2172 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
4 equeuclr 2026 . . . . . 6 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
54sps 2178 . . . . 5 (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
64imim1d 82 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑦𝜑)))
76al2imi 1818 . . . . . 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 1936 . 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 1537  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787
This theorem is referenced by:  ax12  2423  ax12b  2424  equs5  2460  ax12vALT  2469  bj-ax12v3ALT  34868
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