Theorem wsuceq2 31762

Description: Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)

Assertion

Ref	Expression
wsuceq2	⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))

Proof of Theorem wsuceq2

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2622	. 2 ⊢ 𝑋 = 𝑋
3		wsuceq123 31760	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑋) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))
4	1, 2, 3	mp3an13 1415	1 ⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))

Syntax hints:   → wi 4   = wceq 1483  wsuccwsuc 31752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-sup 8348  df-inf 8349  df-wsuc 31756

This theorem is referenced by: (None)