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Theorem drnfc1 2767
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
drnfc1 (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))

Proof of Theorem drnfc1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 drnfc1.1 . . . . 5 (∀𝑥 𝑥 = 𝑦𝐴 = 𝐵)
21eleq2d 2672 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf1 2316 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤𝐴 ↔ Ⅎ𝑦 𝑤𝐵))
43dral2 2311 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑥 𝑤𝐴 ↔ ∀𝑤𝑦 𝑤𝐵))
5 df-nfc 2739 . 2 (𝑥𝐴 ↔ ∀𝑤𝑥 𝑤𝐴)
6 df-nfc 2739 . 2 (𝑦𝐵 ↔ ∀𝑤𝑦 𝑤𝐵)
74, 5, 63bitr4g 301 1 (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472   = wceq 1474  wnf 1698  wcel 1976  wnfc 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-cleq 2602  df-clel 2605  df-nfc 2739
This theorem is referenced by:  nfabd2  2769  nfcvb  4819  nfriotad  6497  bj-nfcsym  31875
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