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Theorem axc11n-16 34542
Description: This theorem shows that, given ax-c16 34496, we can derive a version of ax-c11n 34492. However, it is weaker than ax-c11n 34492 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 34496 . . . 4 (∀𝑥 𝑥 = 𝑧 → (𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
21alrimiv 1895 . . 3 (∀𝑥 𝑥 = 𝑧 → ∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
32axc4i-o 34502 . 2 (∀𝑥 𝑥 = 𝑧 → ∀𝑥𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤))
4 equequ1 1998 . . . . . 6 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
54cbvalv 2309 . . . . . . 7 (∀𝑥 𝑥 = 𝑤 ↔ ∀𝑧 𝑧 = 𝑤)
65a1i 11 . . . . . 6 (𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑤 ↔ ∀𝑧 𝑧 = 𝑤))
74, 6imbi12d 333 . . . . 5 (𝑥 = 𝑧 → ((𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ (𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤)))
87albidv 1889 . . . 4 (𝑥 = 𝑧 → (∀𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ ∀𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤)))
98cbvalv 2309 . . 3 (∀𝑥𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) ↔ ∀𝑧𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
109biimpi 206 . 2 (∀𝑥𝑤(𝑥 = 𝑤 → ∀𝑥 𝑥 = 𝑤) → ∀𝑧𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
11 nfa1-o 34519 . . . . . . 7 𝑧𝑧 𝑧 = 𝑤
121119.23 2118 . . . . . 6 (∀𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) ↔ (∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
1312albii 1787 . . . . 5 (∀𝑤𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) ↔ ∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤))
14 ax6ev 1947 . . . . . . . 8 𝑧 𝑧 = 𝑤
15 pm2.27 42 . . . . . . . 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 1779 . . . . . 6 (∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → ∀𝑤𝑧 𝑧 = 𝑤)
18 equequ2 1999 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑧 = 𝑤𝑧 = 𝑥))
1918spv 2296 . . . . . . . 8 (∀𝑤 𝑧 = 𝑤𝑧 = 𝑥)
2019sps-o 34512 . . . . . . 7 (∀𝑧𝑤 𝑧 = 𝑤𝑧 = 𝑥)
2120alcoms 2075 . . . . . 6 (∀𝑤𝑧 𝑧 = 𝑤𝑧 = 𝑥)
2217, 21syl 17 . . . . 5 (∀𝑤(∃𝑧 𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
2313, 22sylbi 207 . . . 4 (∀𝑤𝑧(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
2423alcoms 2075 . . 3 (∀𝑧𝑤(𝑧 = 𝑤 → ∀𝑧 𝑧 = 𝑤) → 𝑧 = 𝑥)
2524axc4i-o 34502 . 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 196  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-c5 34487  ax-c4 34488  ax-c7 34489  ax-c16 34496
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by: (None)
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