Theorem eqtr2d 2386

Description: An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)

Hypotheses

Ref	Expression
eqtr2d.1	⊢ (φ → A = B)
eqtr2d.2	⊢ (φ → B = C)

Assertion

Ref	Expression
eqtr2d	⊢ (φ → C = A)

Proof of Theorem eqtr2d

Step	Hyp	Ref	Expression
1		eqtr2d.1	. . 3 ⊢ (φ → A = B)
2		eqtr2d.2	. . 3 ⊢ (φ → B = C)
3	1, 2	eqtrd 2385	. 2 ⊢ (φ → A = C)
4	3	eqcomd 2358	1 ⊢ (φ → C = A)

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:  3eqtrrd  2390  3eqtr2rd  2392  ifan  3702  ifor  3703  phi11lem1  4596  enmap2lem3  6066