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Theorem drnfc1 2977
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-11 2159. (Revised by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)
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 2878 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf1 2457 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤𝐴 ↔ Ⅎ𝑦 𝑤𝐵))
43albidv 1921 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑥 𝑤𝐴 ↔ ∀𝑤𝑦 𝑤𝐵))
5 df-nfc 2941 . 2 (𝑥𝐴 ↔ ∀𝑤𝑥 𝑤𝐴)
6 df-nfc 2941 . 2 (𝑦𝐵 ↔ ∀𝑤𝑦 𝑤𝐵)
74, 5, 63bitr4g 317 1 (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wnf 1785  wcel 2112  wnfc 2939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-13 2382  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-cleq 2794  df-clel 2873  df-nfc 2941
This theorem is referenced by:  nfabd2  2981  nfcvb  5245  nfriotad  7108  bj-nfcsym  34340
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