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Theorem drnfc2 2300
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
drnfc2 (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))

Proof of Theorem drnfc2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 drnfc1.1 . . . . 5 (∀𝑥 𝑥 = 𝑦𝐴 = 𝐵)
21eleq2d 2210 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf2 1713 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤𝐴 ↔ Ⅎ𝑧 𝑤𝐵))
43dral2 1710 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑧 𝑤𝐴 ↔ ∀𝑤𝑧 𝑤𝐵))
5 df-nfc 2271 . 2 (𝑧𝐴 ↔ ∀𝑤𝑧 𝑤𝐴)
6 df-nfc 2271 . 2 (𝑧𝐵 ↔ ∀𝑤𝑧 𝑤𝐵)
74, 5, 63bitr4g 222 1 (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1330   = wceq 1332  wnf 1437  wcel 1481  wnfc 2269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136  df-nfc 2271
This theorem is referenced by: (None)
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