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Theorem ax12v2 2046
 Description: It is possible to remove any restriction on 𝜑 in ax12v 2045. Same as Axiom C8 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2016 and ax-13 2245. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
Assertion
Ref Expression
ax12v2 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12v2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equtrr 1946 . . 3 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
2 ax12v 2045 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
31imim1d 82 . . . . 5 (𝑦 = 𝑧 → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑦𝜑)))
43alimdv 1842 . . . 4 (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑧𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
52, 4syl9r 78 . . 3 (𝑦 = 𝑧 → (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
61, 5syld 47 . 2 (𝑦 = 𝑧 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
7 ax6evr 1939 . 2 𝑧 𝑦 = 𝑧
86, 7exlimiiv 1856 1 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  axc16g  2130  axc11rvOLD  2136  sb56  2147  bj-ax12  32276  wl-lem-exsb  32980  wl-lem-moexsb  32982
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