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Theorem bnj1316 31978
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 2970 . . . 4 𝑥𝐴
3 bnj1316.2 . . . . 5 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
43nfcii 2970 . . . 4 𝑥𝐵
52, 4nfeq 2996 . . 3 𝑥 𝐴 = 𝐵
65nf5ri 2187 . 2 (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
76bnj956 31934 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528   = wceq 1530  wcel 2107   ciun 4917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rex 3149  df-iun 4919
This theorem is referenced by:  bnj1000  32099  bnj1318  32181
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