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Theorem axc16g-o 36192
Description: A generalization of axiom ax-c16 36150. Version of axc16g 2262 using ax-c11 36145. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc16g-o (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16g-o
StepHypRef Expression
1 aev-o 36189 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
2 ax-c16 36150 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
3 biidd 265 . . . 4 (∀𝑧 𝑧 = 𝑥 → (𝜑𝜑))
43dral1-o 36162 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑))
54biimprd 251 . 2 (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑))
61, 2, 5sylsyld 61 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2161  ax-c5 36141  ax-c4 36142  ax-c7 36143  ax-c10 36144  ax-c11 36145  ax-c9 36148  ax-c16 36150
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  ax12inda2  36205
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