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Theorem axc16gb 2284
Description: Biconditional strengthening of axc16g 2282. (Contributed by NM, 15-May-1993.)
Assertion
Ref Expression
axc16gb (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem axc16gb
StepHypRef Expression
1 axc16g 2282 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
2 sp 2217 . 2 (∀𝑧𝜑𝜑)
31, 2impbid1 217 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876
This theorem is referenced by:  sbal  2582
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