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Theorem axc11nlemALT 2310
Description: Alternate version of axc11nlemOLD2 1990 used in an older proof. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11nlemALT (∀𝑥 𝑥 = 𝑤 → ∀𝑦 𝑦 = 𝑥)
Distinct variable groups:   𝑥,𝑤   𝑦,𝑤

Proof of Theorem axc11nlemALT
StepHypRef Expression
1 cbvaev 1981 . . 3 (∀𝑥 𝑥 = 𝑤 → ∀𝑦 𝑦 = 𝑤)
2 equequ2 1955 . . . . 5 (𝑥 = 𝑤 → (𝑦 = 𝑥𝑦 = 𝑤))
32biimprd 238 . . . 4 (𝑥 = 𝑤 → (𝑦 = 𝑤𝑦 = 𝑥))
43al2imi 1740 . . 3 (∀𝑦 𝑥 = 𝑤 → (∀𝑦 𝑦 = 𝑤 → ∀𝑦 𝑦 = 𝑥))
51, 4syl5com 31 . 2 (∀𝑥 𝑥 = 𝑤 → (∀𝑦 𝑥 = 𝑤 → ∀𝑦 𝑦 = 𝑥))
6 dveeq1 2304 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑤 → ∀𝑦 𝑥 = 𝑤))
76spsd 2060 . . . 4 (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑥 𝑥 = 𝑤 → ∀𝑦 𝑥 = 𝑤))
87com12 32 . . 3 (∀𝑥 𝑥 = 𝑤 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑤))
98con1d 139 . 2 (∀𝑥 𝑥 = 𝑤 → (¬ ∀𝑦 𝑥 = 𝑤 → ∀𝑦 𝑦 = 𝑥))
105, 9pm2.61d 170 1 (∀𝑥 𝑥 = 𝑤 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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-10 2021  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  axc11nALT  2314  aevlemALTOLD  2324
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