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Theorem bnj551 34282
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj551 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)

Proof of Theorem bnj551
StepHypRef Expression
1 eqtr2 2750 . 2 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → suc 𝑝 = suc 𝑖)
2 suc11reg 9613 . 2 (suc 𝑝 = suc 𝑖𝑝 = 𝑖)
31, 2sylib 217 1 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  suc csuc 6359
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721  ax-reg 9586
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-eprel 5573  df-fr 5624  df-suc 6363
This theorem is referenced by:  bnj554  34439  bnj557  34441  bnj966  34484
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