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Theorem drnfc2 2997
 Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1915 with dral2 2454, leading to a one byte longer proof. However feel free to manually edit it according to conventions. Usage of this theorem is discouraged because it depends on ax-13 2384. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
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 2896 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
32drnf2 2460 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤𝐴 ↔ Ⅎ𝑧 𝑤𝐵))
43albidv 1915 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑧 𝑤𝐴 ↔ ∀𝑤𝑧 𝑤𝐵))
5 df-nfc 2961 . 2 (𝑧𝐴 ↔ ∀𝑤𝑧 𝑤𝐴)
6 df-nfc 2961 . 2 (𝑧𝐵 ↔ ∀𝑤𝑧 𝑤𝐵)
74, 5, 63bitr4g 316 1 (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1529   = wceq 1531  Ⅎwnf 1778   ∈ wcel 2108  Ⅎwnfc 2959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-13 2384  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-cleq 2812  df-clel 2891  df-nfc 2961 This theorem is referenced by: (None)
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