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Theorem ax12vOLDOLD 2048
 Description: Obsolete proof of ax12v 2045 as of 7-Mar-2021. (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.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax12vOLDOLD (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12vOLDOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1950 . . . 4 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
21biimprd 238 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
3 ax-5 1836 . . . . 5 (𝜑 → ∀𝑧𝜑)
4 ax-12 2044 . . . . 5 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
53, 4syl5 34 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
62imim1d 82 . . . . 5 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑦𝜑)))
76alimdv 1842 . . . 4 (𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
85, 7syl9r 78 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
92, 8syld 47 . 2 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
10 ax6ev 1887 . 2 𝑧 𝑧 = 𝑦
119, 10exlimiiv 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: (None)
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