Theorem imaeq12d 4944

Description: Equality theorem for image. (Contributed by SF, 8-Jan-2018.)

Hypotheses

Ref	Expression
imaeq1d.1	⊢ (φ → A = B)
imaeq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
imaeq12d	⊢ (φ → (A “ C) = (B “ D))

Proof of Theorem imaeq12d

Step	Hyp	Ref	Expression
1		imaeq1d.1	. . 3 ⊢ (φ → A = B)
2	1	imaeq1d 4942	. 2 ⊢ (φ → (A “ C) = (B “ C))
3		imaeq12d.2	. . 3 ⊢ (φ → C = D)
4	3	imaeq2d 4943	. 2 ⊢ (φ → (B “ C) = (B “ D))
5	2, 4	eqtrd 2385	1 ⊢ (φ → (A “ C) = (B “ D))

Syntax hints:   → wi 4   = wceq 1642   “ cima 4723

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-or 359  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-nfc 2479  df-rex 2621  df-br 4641  df-ima 4728

This theorem is referenced by:  csbima12g  4956