Theorem bnj551 35239

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 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖)

Proof of Theorem bnj551

Step	Hyp	Ref	Expression
1		eqtr2 2783	. 2 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → suc 𝑝 = suc 𝑖)
2		suc11reg 9601	. 2 ⊢ (suc 𝑝 = suc 𝑖 ↔ 𝑝 = 𝑖)
3	1, 2	sylib 221	1 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  suc csuc 6363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7739  ax-reg 9567

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-eprel 5559  df-fr 5612  df-suc 6367

This theorem is used by:  bnj554  35395  bnj557  35397  bnj966  35440