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Theorem nfcjust 2739
 Description: Justification theorem for df-nfc 2740. (Contributed by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
nfcjust (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfcjust
StepHypRef Expression
1 nfv 1830 . . 3 𝑥 𝑦 = 𝑧
2 eleq1 2676 . . 3 (𝑦 = 𝑧 → (𝑦𝐴𝑧𝐴))
31, 2nfbidf 2079 . 2 (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑧𝐴))
43cbvalv 2261 1 (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-cleq 2603  df-clel 2606 This theorem is referenced by: (None)
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