Theorem 3eqtr4a 2411

Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
3eqtr4a.1	⊢ A = B
3eqtr4a.2	⊢ (φ → C = A)
3eqtr4a.3	⊢ (φ → D = B)

Assertion

Ref	Expression
3eqtr4a	⊢ (φ → C = D)

Proof of Theorem 3eqtr4a

Step	Hyp	Ref	Expression
1		3eqtr4a.2	. . 3 ⊢ (φ → C = A)
2		3eqtr4a.1	. . 3 ⊢ A = B
3	1, 2	syl6eq 2401	. 2 ⊢ (φ → C = B)
4		3eqtr4a.3	. 2 ⊢ (φ → D = B)
5	3, 4	eqtr4d 2388	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:  uniintsn  3964  iununi  4051  pw1eqadj  4333  tfincl  4493  dmxpid  4925  imasn  5019  rnxpid  5055  1st2nd2  5517  uniqs2  5986  muccom  6135  mucass  6136  mucid1  6144  tcdi  6165  tce2  6237  nchoicelem2  6291