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Theorem bnj551 32006
 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 2840 . 2 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → suc 𝑝 = suc 𝑖)
2 suc11reg 9074 . 2 (suc 𝑝 = suc 𝑖𝑝 = 𝑖)
31, 2sylib 220 1 ((𝑚 = suc 𝑝𝑚 = suc 𝑖) → 𝑝 = 𝑖)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531  suc csuc 6186 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453  ax-reg 9048 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-eprel 5458  df-fr 5507  df-suc 6190 This theorem is referenced by:  bnj554  32164  bnj557  32166  bnj966  32209
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