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Theorem drnf2 1664
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
drex2.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
drnf2 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Proof of Theorem drnf2
StepHypRef Expression
1 drex2.1 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
21dral2 1661 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
31, 2imbi12d 232 . . 3 (∀𝑥 𝑥 = 𝑦 → ((𝜑 → ∀𝑧𝜑) ↔ (𝜓 → ∀𝑧𝜓)))
43dral2 1661 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝜑 → ∀𝑧𝜑) ↔ ∀𝑧(𝜓 → ∀𝑧𝜓)))
5 df-nf 1391 . 2 (Ⅎ𝑧𝜑 ↔ ∀𝑧(𝜑 → ∀𝑧𝜑))
6 df-nf 1391 . 2 (Ⅎ𝑧𝜓 ↔ ∀𝑧(𝜓 → ∀𝑧𝜓))
74, 5, 63bitr4g 221 1 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  nfsbxy  1861  nfsbxyt  1862  drnfc2  2240
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