Theorem cnvkeqd 4218

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

Hypothesis

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

Assertion

Ref	Expression
cnvkeqd	⊢ (φ → ◡kA = ◡kB)

Proof of Theorem cnvkeqd

Step	Hyp	Ref	Expression
1		cnvkeqd.1	. 2 ⊢ (φ → A = B)
2		cnvkeq 4216	. 2 ⊢ (A = B → ◡kA = ◡kB)
3	1, 2	syl 15	1 ⊢ (φ → ◡kA = ◡kB)

Syntax hints:   → wi 4   = wceq 1642  ^◡_kccnvk 4176

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-cnvk 4187

This theorem is referenced by:  imagekeq  4245