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Theorem axc11n-16 33044
Description: This theorem shows that, given ax-c16 32998, we can derive a version of ax-c11n 32994. However, it is weaker than ax-c11n 32994 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11n-16 (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
Distinct variable group:   𝑥,𝑧

Proof of Theorem axc11n-16
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-c16 32998 . . . 4 (∀𝑥 𝑥 = 𝑧 → (𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
21alrimiv 1841 . . 3 (∀𝑥 𝑥 = 𝑧 → ∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
32axc4i-o 33004 . 2 (∀𝑥 𝑥 = 𝑧 → ∀𝑥𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
4 equequ1 1938 . . . . . 6 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
54cbvalv 2259 . . . . . . 7 (∀𝑥 𝑥 = 𝑤 ↔ ∀𝑧 𝑧 = 𝑤)
65a1i 11 . . . . . 6 (𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑤 ↔ ∀𝑧 𝑧 = 𝑤))
74, 6imbi12d 332 . . . . 5 (𝑥 = 𝑧 → ((𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ (𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤)))
87albidv 1835 . . . 4 (𝑥 = 𝑧 → (∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ ∀𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤)))
98cbvalv 2259 . . 3 (∀𝑥𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ ∀𝑧𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
109biimpi 204 . 2 (∀𝑥𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) → ∀𝑧𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
11 nfa1-o 33021 . . . . . . 7 𝑧𝑧 𝑧 = 𝑤
121119.23 2065 . . . . . 6 (∀𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) ↔ (∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
1312albii 1736 . . . . 5 (∀𝑤𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) ↔ ∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
14 ax6ev 1876 . . . . . . . 8 𝑧 𝑧 = 𝑤
15 pm2.27 40 . . . . . . . 8 (∃𝑧 𝑧 = 𝑤 → ((∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑧 𝑧 = 𝑤))
1614, 15ax-mp 5 . . . . . . 7 ((∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑧 𝑧 = 𝑤)
1716alimi 1729 . . . . . 6 (∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑤𝑧 𝑧 = 𝑤)
18 equequ2 1939 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑧 = 𝑤𝑧 = 𝑥))
1918spv 2246 . . . . . . . 8 (∀𝑤 𝑧 = 𝑤𝑧 = 𝑥)
2019sps-o 33014 . . . . . . 7 (∀𝑧𝑤 𝑧 = 𝑤𝑧 = 𝑥)
2120alcoms 2021 . . . . . 6 (∀𝑤𝑧 𝑧 = 𝑤𝑧 = 𝑥)
2217, 21syl 17 . . . . 5 (∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
2313, 22sylbi 205 . . . 4 (∀𝑤𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
2423alcoms 2021 . . 3 (∀𝑧𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
2524axc4i-o 33004 . 2 (∀𝑧𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑧 𝑧 = 𝑥)
263, 10, 253syl 18 1 (∀𝑥 𝑥 = 𝑧 → ∀𝑧 𝑧 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-c5 32989  ax-c4 32990  ax-c7 32991  ax-c16 32998
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ex 1695  df-nf 1700
This theorem is referenced by: (None)
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