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Theorem axc15OLD 2436
Description: Obsolete proof of axc15 2435 as of 26-Mar-2023. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc15OLD (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Proof of Theorem axc15OLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1963 . 2 𝑧 𝑧 = 𝑦
2 dveeq2 2387 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
3 ax12v 2168 . . . . 5 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
4 equequ2 2024 . . . . . . 7 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
54sps 2174 . . . . . 6 (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
6 nfa1 2146 . . . . . . . 8 𝑥𝑥 𝑧 = 𝑦
75imbi1d 343 . . . . . . . 8 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
86, 7albid 2214 . . . . . . 7 (∀𝑥 𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
98imbi2d 342 . . . . . 6 (∀𝑥 𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
105, 9imbi12d 346 . . . . 5 (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
113, 10mpbii 234 . . . 4 (∀𝑥 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
122, 11syl6 35 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
1312exlimdv 1925 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
141, 13mpi 20 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776
This theorem is referenced by: (None)
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