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Theorem drnfc2 2812
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypothesis
Ref Expression
drnfc1.1 (∀𝑥 𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
drnfc2 (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))

Proof of Theorem drnfc2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 drnfc1.1 . . . . 5 (∀𝑥 𝑥 = 𝑦𝐴 = 𝐵)
21eleq2d 2716 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf2 2361 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤𝐴 ↔ Ⅎ𝑧 𝑤𝐵))
43dral2 2355 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑧 𝑤𝐴 ↔ ∀𝑤𝑧 𝑤𝐵))
5 df-nfc 2782 . 2 (𝑧𝐴 ↔ ∀𝑤𝑧 𝑤𝐴)
6 df-nfc 2782 . 2 (𝑧𝐵 ↔ ∀𝑤𝑧 𝑤𝐵)
74, 5, 63bitr4g 303 1 (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521   = wceq 1523  wnf 1748  wcel 2030  wnfc 2780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-cleq 2644  df-clel 2647  df-nfc 2782
This theorem is referenced by: (None)
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