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Theorem bnj1316 29939
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1316.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
bnj1316.2 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
Assertion
Ref Expression
bnj1316 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem bnj1316
StepHypRef Expression
1 bnj1316.1 . . . . 5 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
21nfcii 2742 . . . 4 𝑥𝐴
3 bnj1316.2 . . . . 5 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
43nfcii 2742 . . . 4 𝑥𝐵
52, 4nfeq 2762 . . 3 𝑥 𝐴 = 𝐵
65nf5ri 2053 . 2 (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
76bnj956 29895 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473   = wceq 1475  wcel 1977   ciun 4450
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-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-iun 4452
This theorem is referenced by:  bnj1000  30059  bnj1318  30141
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