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Theorem axc16nfOLD 2199
 Description: Obsolete proof of axc16nf 2175 as of 12-Oct-2021. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2074. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
axc16nfOLD (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16nfOLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 aev 2025 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑤)
2 nfa1 2068 . . 3 𝑧𝑧 𝑧 = 𝑤
3 axc16 2173 . . 3 (∀𝑧 𝑧 = 𝑤 → (𝜑 → ∀𝑧𝜑))
42, 3nf5d 2156 . 2 (∀𝑧 𝑧 = 𝑤 → Ⅎ𝑧𝜑)
51, 4syl 17 1 (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521  Ⅎwnf 1748 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-12 2087 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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