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Theorem dral1 1618
 Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)
Hypothesis
Ref Expression
dral1.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dral1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Proof of Theorem dral1
StepHypRef Expression
1 hbae 1606 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑥 𝑥 = 𝑦)
2 dral1.1 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 132 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
41, 3alimdh 1356 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑥𝜓))
5 ax10o 1603 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓))
64, 5syld 40 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜓))
7 hbae 1606 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑦𝑥 𝑥 = 𝑦)
82biimprd 147 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜓𝜑))
97, 8alimdh 1356 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑦𝜑))
10 ax10o 1603 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∀𝑦𝜑 → ∀𝑥𝜑))
1110alequcoms 1409 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
129, 11syld 40 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜑))
136, 12impbid 120 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀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-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 This theorem is referenced by:  drnf1  1621  equveli  1642  a16g  1744
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