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Theorem axc16g-o 38935
Description: A generalization of Axiom ax-c16 38893. Version of axc16g 2260 using ax-c11 38888. (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 38932 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
2 ax-c16 38893 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
3 biidd 262 . . . 4 (∀𝑧 𝑧 = 𝑥 → (𝜑𝜑))
43dral1-o 38905 . . 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 1538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-11 2157  ax-c5 38884  ax-c4 38885  ax-c7 38886  ax-c10 38887  ax-c11 38888  ax-c9 38891  ax-c16 38893
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780
This theorem is referenced by:  ax12inda2  38948
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