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Theorem dral2-o 36126
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral2 2462 using ax-c11 36083. (Contributed by NM, 27-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dral2-o.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dral2-o (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Proof of Theorem dral2-o
StepHypRef Expression
1 hbae-o 36099 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
2 dral2-o.1 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albidh 1868 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-11 2162  ax-c5 36079  ax-c4 36080  ax-c7 36081  ax-c10 36082  ax-c11 36083  ax-c9 36086
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  ax12eq  36137  ax12el  36138  ax12indalem  36141  ax12inda2ALT  36142
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