Theorem 3eqtr3a 2409

Description: A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)

Hypotheses

Ref	Expression
3eqtr3a.1	⊢ A = B
3eqtr3a.2	⊢ (φ → A = C)
3eqtr3a.3	⊢ (φ → B = D)

Assertion

Ref	Expression
3eqtr3a	⊢ (φ → C = D)

Proof of Theorem 3eqtr3a

Step	Hyp	Ref	Expression
1		3eqtr3a.2	. 2 ⊢ (φ → A = C)
2		3eqtr3a.1	. . 3 ⊢ A = B
3		3eqtr3a.3	. . 3 ⊢ (φ → B = D)
4	2, 3	syl5eq 2397	. 2 ⊢ (φ → A = D)
5	1, 4	eqtr3d 2387	1 ⊢ (φ → C = D)

Syntax hints:   → wi 4   = wceq 1642

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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346

This theorem is referenced by:  uneqin  3506