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Theorem ax12vOLD 2052
Description: Obsolete proof of ax12v2 2051 as of 24-Mar-2021. (Contributed by NM, 5-Aug-1993.) Removed dependencies on ax-10 2021 and ax-13 2250. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.) (Proof shortened by Wolf Lammen, 7-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax12vOLD (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12vOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equtrr 1951 . . 3 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
2 ax-5 1841 . . . . 5 (𝜑 → ∀𝑧𝜑)
3 ax-12 2049 . . . . 5 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
42, 3syl5 34 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
51imim1d 82 . . . . 5 (𝑦 = 𝑧 → ((𝑥 = 𝑧𝜑) → (𝑥 = 𝑦𝜑)))
65alimdv 1847 . . . 4 (𝑦 = 𝑧 → (∀𝑥(𝑥 = 𝑧𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
74, 6syl9r 78 . . 3 (𝑦 = 𝑧 → (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
81, 7syld 47 . 2 (𝑦 = 𝑧 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
9 ax6evr 1944 . 2 𝑧 𝑦 = 𝑧
108, 9exlimiiv 1861 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 1841  ax-6 1890  ax-7 1937  ax-12 2049
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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