Theorem wsuceq3 35781

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

Assertion

Ref	Expression
wsuceq3	⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))

Proof of Theorem wsuceq3

Step	Hyp	Ref	Expression
1		eqid 2740	. 2 ⊢ 𝑅 = 𝑅
2		eqid 2740	. 2 ⊢ 𝐴 = 𝐴
3		wsuceq123 35778	. 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))
4	1, 2, 3	mp3an12 1451	1 ⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))

Syntax hints:   → wi 4   = wceq 1537  wsuccwsuc 35774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-sup 9511  df-inf 9512  df-wsuc 35776

This theorem is referenced by: (None)