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Theorem a16g 1744
Description: A generalization of axiom ax-16 1695. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
a16g (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem a16g
StepHypRef Expression
1 aev 1693 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
2 ax16 1694 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
3 biidd 161 . . . 4 (∀𝑧 𝑧 = 𝑥 → (𝜑𝜑))
43dral1 1618 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑))
54biimprd 147 . 2 (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑))
61, 2, 5sylsyld 52 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646
This theorem is referenced by:  a16gb  1745  a16nf  1746
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