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Theorem drnfc1 2194
 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 2107 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf1 1621 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤𝐴 ↔ Ⅎ𝑦 𝑤𝐵))
43dral2 1619 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑥 𝑤𝐴 ↔ ∀𝑤𝑦 𝑤𝐵))
5 df-nfc 2167 . 2 (𝑥𝐴 ↔ ∀𝑤𝑥 𝑤𝐴)
6 df-nfc 2167 . 2 (𝑦𝐵 ↔ ∀𝑤𝑦 𝑤𝐵)
74, 5, 63bitr4g 212 1 (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  Ⅎwnfc 2165 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167 This theorem is referenced by: (None)
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