Theorem sylan9eqOLD 1949

Description: An equality transitivity deduction.

Hypotheses

Ref	Expression
sylan9eq.1	|- (ph -> A = B)
sylan9eq.2	|- (ps -> B = C)

Assertion

Ref	Expression
sylan9eqOLD	|- ((ph /\ ps) -> A = C)

Proof of Theorem sylan9eqOLD

Step	Hyp	Ref	Expression
1		sylan9eq.1	. . 3 |- (ph -> A = B)
2	1	adantr 425	. 2 |- ((ph /\ ps) -> A = B)
3		sylan9eq.2	. . 3 |- (ps -> B = C)
4	3	adantl 424	. 2 |- ((ph /\ ps) -> B = C)
5	2, 4	eqtrd 1925	1 |- ((ph /\ ps) -> A = C)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-cleq 1877

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