Theorem 3eqtr3g 2408

Description: A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)

Hypotheses

Ref	Expression
3eqtr3g.1	⊢ (φ → A = B)
3eqtr3g.2	⊢ A = C
3eqtr3g.3	⊢ B = D

Assertion

Ref	Expression
3eqtr3g	⊢ (φ → C = D)

Proof of Theorem 3eqtr3g

Step	Hyp	Ref	Expression
1		3eqtr3g.2	. . 3 ⊢ A = C
2		3eqtr3g.1	. . 3 ⊢ (φ → A = B)
3	1, 2	syl5eqr 2399	. 2 ⊢ (φ → C = B)
4		3eqtr3g.3	. 2 ⊢ B = D
5	3, 4	syl6eq 2401	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:  csbnest1g  3189  nineq2  3236  compleqb  3544  adj11  3890  pw1eqadj  4333  tfindi  4497  opth  4603  xpid11  4927  cnveqb  5064  cores2  5092  fvunsn  5445  nchoicelem1  6290