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Theorem drnfc1 2336
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 2247 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf1 1733 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤𝐴 ↔ Ⅎ𝑦 𝑤𝐵))
43dral2 1731 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑥 𝑤𝐴 ↔ ∀𝑤𝑦 𝑤𝐵))
5 df-nfc 2308 . 2 (𝑥𝐴 ↔ ∀𝑤𝑥 𝑤𝐴)
6 df-nfc 2308 . 2 (𝑦𝐵 ↔ ∀𝑤𝑦 𝑤𝐵)
74, 5, 63bitr4g 223 1 (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353  wnf 1460  wcel 2148  wnfc 2306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-nfc 2308
This theorem is referenced by: (None)
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