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Theorem bnj1316 31219
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 2893 . . . 4 𝑥𝐴
3 bnj1316.2 . . . . 5 (𝑦𝐵 → ∀𝑥 𝑦𝐵)
43nfcii 2893 . . . 4 𝑥𝐵
52, 4nfeq 2914 . . 3 𝑥 𝐴 = 𝐵
65nf5ri 2212 . 2 (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵)
76bnj956 31175 1 (𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630   = wceq 1632  wcel 2139   ciun 4672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-iun 4674
This theorem is referenced by:  bnj1000  31339  bnj1318  31421
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