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Theorem drnfc2 2885
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 2789 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf2 2434 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤𝐴 ↔ Ⅎ𝑧 𝑤𝐵))
43dral2 2428 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑧 𝑤𝐴 ↔ ∀𝑤𝑧 𝑤𝐵))
5 df-nfc 2855 . 2 (𝑧𝐴 ↔ ∀𝑤𝑧 𝑤𝐴)
6 df-nfc 2855 . 2 (𝑧𝐵 ↔ ∀𝑤𝑧 𝑤𝐵)
74, 5, 63bitr4g 303 1 (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1594   = wceq 1596  wnf 1821  wcel 2103  wnfc 2853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-cleq 2717  df-clel 2720  df-nfc 2855
This theorem is referenced by: (None)
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