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Theorem drnfc1 2768
 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 2673 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf1 2317 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤𝐴 ↔ Ⅎ𝑦 𝑤𝐵))
43dral2 2312 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑥 𝑤𝐴 ↔ ∀𝑤𝑦 𝑤𝐵))
5 df-nfc 2740 . 2 (𝑥𝐴 ↔ ∀𝑤𝑥 𝑤𝐴)
6 df-nfc 2740 . 2 (𝑦𝐵 ↔ ∀𝑤𝑦 𝑤𝐵)
74, 5, 63bitr4g 302 1 (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606  df-nfc 2740 This theorem is referenced by:  nfabd2  2770  nfcvb  4824  nfriotad  6519  bj-nfcsym  32079
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