Theorem 3eqtrd 2389

Description: A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)

Hypotheses

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

Assertion

Ref	Expression
3eqtrd	⊢ (φ → A = D)

Proof of Theorem 3eqtrd

Step	Hyp	Ref	Expression
1		3eqtrd.1	. 2 ⊢ (φ → A = B)
2		3eqtrd.2	. . 3 ⊢ (φ → B = C)
3		3eqtrd.3	. . 3 ⊢ (φ → C = D)
4	2, 3	eqtrd 2385	. 2 ⊢ (φ → B = D)
5	1, 4	eqtrd 2385	1 ⊢ (φ → A = 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:  tpeq123d  3815  diftpsn3  3850  vfinncsp  4555  fvun  5379  oveq123d  5544  fvmptd  5703