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Theorem dral2-o 37442
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 2437 using ax-c11 37399. (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 37415 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
2 dral2-o.1 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2albidh 1870 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-11 2155  ax-c5 37395  ax-c4 37396  ax-c7 37397  ax-c10 37398  ax-c11 37399  ax-c9 37402
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783
This theorem is referenced by:  ax12eq  37453  ax12el  37454  ax12indalem  37457  ax12inda2ALT  37458
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