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Theorem axc11nfromc11 35942
Description: Rederivation of ax-c11n 35904 from original version ax-c11 35903. See theorem axc11 2444 for the derivation of ax-c11 35903 from ax-c11n 35904.

This theorem should not be referenced in any proof. Instead, use ax-c11n 35904 above so that uses of ax-c11n 35904 can be more easily identified, or use aecom-o 35917 when this form is needed for studies involving ax-c11 35903 and omitting ax-5 1902. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
axc11nfromc11 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem axc11nfromc11
StepHypRef Expression
1 ax-c11 35903 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi 2015 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1803 . 2 (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
52, 4syl 17 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526
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-c11 35903
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by: (None)
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