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Theorem bnj1316 34834
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 2894 . . . 4 𝑥𝐴
3 bnj1316.2 . . . . 5 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
43nfcii 2894 . . . 4 𝑥𝐵
52, 4nfeq 2919 . . 3 𝑥 𝐴 = 𝐵
65nf5ri 2195 . 2 (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
76bnj956 34790 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wcel 2108   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rex 3071  df-iun 4993
This theorem is referenced by:  bnj1000  34955  bnj1318  35039
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