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Theorem dveeq1-o16 35939
Description: Version of dveeq1 2393 using ax-c16 35895 instead of ax-5 1904. (Contributed by NM, 29-Apr-2008.) TODO: Recover proof from older set.mm to remove use of ax-5 1904. (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dveeq1-o16 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Distinct variable group:   𝑥,𝑧

Proof of Theorem dveeq1-o16
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax5eq 35935 . 2 (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧)
2 ax5eq 35935 . 2 (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
3 equequ1 2025 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
41, 2, 3dvelimh 2469 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-c9 35893  ax-c16 35895
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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