Theorem 3eqtr4rd 1135

Description: A deduction from three chained equalities.

Hypotheses

Ref	Expression
3eqtr4d.1	⊢ (φ → A = B)
3eqtr4d.2	⊢ (φ → C = A)
3eqtr4d.3	⊢ (φ → D = B)

Assertion

Ref	Expression
3eqtr4rd	⊢ (φ → D = C)

Proof of Theorem 3eqtr4rd

Step	Hyp	Ref	Expression
1		3eqtr4d.3	. . 3 ⊢ (φ → D = B)
2		3eqtr4d.1	. . 3 ⊢ (φ → A = B)
3	1, 2	eqtr4d 1131	. 2 ⊢ (φ → D = A)
4		3eqtr4d.2	. 2 ⊢ (φ → C = A)
5	3, 4	eqtr4d 1131	1 ⊢ (φ → D = C)

Syntax hints:   → wi 2   = wceq 1091

This theorem is referenced by:  nndi 3180  phplem5 3407  divnegt 4259  hvsubidt 5005  pjtot 5644

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097

metamath.org