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Theorem bnj551 31119
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 2780 . 2 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → suc 𝑝 = suc 𝑖)
2 suc11reg 8689 . 2 (suc 𝑝 = suc 𝑖𝑝 = 𝑖)
31, 2sylib 208 1 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  suc csuc 5886
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-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114  ax-reg 8662
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-eprel 5179  df-fr 5225  df-suc 5890
This theorem is referenced by:  bnj554  31276  bnj557  31278  bnj966  31321
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