Theorem predeq2 6153

Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

Ref	Expression
predeq2	⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Proof of Theorem predeq2

Step	Hyp	Ref	Expression
1		eqid 2823	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2823	. 2 ⊢ 𝑋 = 𝑋
3		predeq123 6151	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
4	1, 2, 3	mp3an13 1448	1 ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   = wceq 1537  Predcpred 6149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150

This theorem is referenced by:  wrecseq123  7950  wfrlem5  7961  prednn  13033  prednn0  13034  trpredeq2  33062  frmin  33086  fprlem1  33139  frrlem15  33144