Theorem eqeqan12dOLD 1902

Description: A useful inference for substituting definitions into an equality.

Hypotheses

Ref	Expression
eqeqan12d.1	|- (ph -> A = B)
eqeqan12d.2	|- (ps -> C = D)

Assertion

Ref	Expression
eqeqan12dOLD	|- ((ph /\ ps) -> (A = C <-> B = D))

Proof of Theorem eqeqan12dOLD

Step	Hyp	Ref	Expression
1		eqeqan12d.1	. . 3 |- (ph -> A = B)
2	1	adantr 425	. 2 |- ((ph /\ ps) -> A = B)
3		eqeqan12d.2	. . 3 |- (ps -> C = D)
4	3	adantl 424	. 2 |- ((ph /\ ps) -> C = D)
5	2, 4	eqeq12d 1899	1 |- ((ph /\ ps) -> (A = C <-> B = D))

Syntax hints:   -> wi 3   <-> wb 163   /\ 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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