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Theorem bnj1316 31342
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 2898 . . . 4 𝑥𝐴
3 bnj1316.2 . . . . 5 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
43nfcii 2898 . . . 4 𝑥𝐵
52, 4nfeq 2919 . . 3 𝑥 𝐴 = 𝐵
65nf5ri 2227 . 2 (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
76bnj956 31298 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650   = wceq 1652  wcel 2155   ciun 4678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-iun 4680
This theorem is referenced by:  bnj1000  31462  bnj1318  31544
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