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Theorem nfcjust 2904
Description: Justification theorem for df-nfc 2905. (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 1998 . . 3 𝑥 𝑦 = 𝑧
2 eleq1w 2836 . . 3 (𝑦 = 𝑧 → (𝑦𝐴𝑧𝐴))
31, 2nfbidf 2251 . 2 (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦𝐴 ↔ Ⅎ𝑥 𝑧𝐴))
43cbvalvw 2128 1 (∀𝑦𝑥 𝑦𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1632  wnf 1859  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-12 2206
This theorem depends on definitions:  df-bi 198  df-an 384  df-ex 1856  df-nf 1861  df-clel 2770
This theorem is referenced by: (None)
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