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Theorem a16gb 1852
Description: A generalization of Axiom ax-16 1801. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
a16gb (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem a16gb
StepHypRef Expression
1 a16g 1851 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
2 ax-4 1497 . 2 (∀𝑧𝜑𝜑)
31, 2impbid1 141 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750
This theorem is referenced by: (None)
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