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Theorem axc16g-o 38916
Description: A generalization of Axiom ax-c16 38874. Version of axc16g 2258 using ax-c11 38869. (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 38913 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
2 ax-c16 38874 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
3 biidd 262 . . . 4 (∀𝑧 𝑧 = 𝑥 → (𝜑𝜑))
43dral1-o 38886 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑))
54biimprd 248 . 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 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-11 2155  ax-c5 38865  ax-c4 38866  ax-c7 38867  ax-c10 38868  ax-c11 38869  ax-c9 38872  ax-c16 38874
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777
This theorem is referenced by:  ax12inda2  38929
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