Theorem ins3keqd 4223

Description: Equality deduction for Kuratowski insert three operator. (Contributed by SF, 12-Jan-2015.)

Hypothesis

Ref	Expression
inskeqd.1	⊢ (φ → A = B)

Assertion

Ref	Expression
ins3keqd	⊢ (φ → Ins3k A = Ins3k B)

Proof of Theorem ins3keqd

Step	Hyp	Ref	Expression
1		inskeqd.1	. 2 ⊢ (φ → A = B)
2		ins3keq 4219	. 2 ⊢ (A = B → Ins3k A = Ins3k B)
3	1, 2	syl 15	1 ⊢ (φ → Ins3k A = Ins3k B)

Syntax hints:   → wi 4   = wceq 1642   Ins3_k cins3k 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ins3k 4188

This theorem is referenced by:  cokeq2  4231  imagekeq  4244