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Theorem dral1-o 33708
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 dral1 2324 using ax-c11 33691. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.)
Hypothesis
Ref Expression
dral1-o.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dral1-o (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Proof of Theorem dral1-o
StepHypRef Expression
1 hbae-o 33707 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
2 dral1-o.1 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 219 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
41, 3alimdh 1742 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑥𝜓))
5 ax-c11 33691 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
64, 5syld 47 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜓))
7 hbae-o 33707 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
82biimprd 238 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜓𝜑))
97, 8alimdh 1742 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑦𝜑))
10 ax-c11 33691 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∀𝑦𝜑 → ∀𝑥𝜑))
1110aecoms-o 33706 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
129, 11syld 47 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜑))
136, 12impbid 202 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-11 2031  ax-c5 33687  ax-c4 33688  ax-c7 33689  ax-c10 33690  ax-c11 33691  ax-c9 33694
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  ax12fromc15  33709  axc16g-o  33738  ax12indalem  33749  ax12inda2ALT  33750
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