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Theorem drnfc2 2914
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1916 with dral2 2432, leading to a one byte longer proof. However feel free to manually edit it according to conventions. (TODO: dral2 2432 depends on ax-13 2366, hence its usage during minimizing is discouraged. Check in the long run whether this is a permanent restriction). Usage of this theorem is discouraged because it depends on ax-13 2366. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2101. (Revised by Wolf Lammen, 22-Sep-2024.) (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)
2 eleq2w2 2722 . . . . 5 (𝐴 = 𝐵 → (𝑤𝐴𝑤𝐵))
31, 2syl 17 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑤𝐴𝑤𝐵))
43drnf2 2438 . . 3 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤𝐴 ↔ Ⅎ𝑧 𝑤𝐵))
54albidv 1916 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑤𝑧 𝑤𝐴 ↔ ∀𝑤𝑧 𝑤𝐵))
6 df-nfc 2878 . 2 (𝑧𝐴 ↔ ∀𝑤𝑧 𝑤𝐴)
7 df-nfc 2878 . 2 (𝑧𝐵 ↔ ∀𝑤𝑧 𝑤𝐵)
85, 6, 73bitr4g 313 1 (∀𝑥 𝑥 = 𝑦 → (𝑧𝐴𝑧𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532   = wceq 1534  wnf 1778  wcel 2099  wnfc 2876
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 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2366  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-cleq 2718  df-nfc 2878
This theorem is referenced by: (None)
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