eqeq12 - New Foundations Explorer

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Theorem eqeq12 2365

Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)

**Assertion**
Ref	Expression
eqeq12	⊢ ((A = B ∧ C = D) → (A = C ↔ B = D))

**Proof of Theorem eqeq12**
Step	Hyp	Ref	Expression
1		eqeq1 2359	. 2 ⊢ (A = B → (A = C ↔ B = C))
2		eqeq2 2362	. 2 ⊢ (C = D → (B = C ↔ B = D))
3	1, 2	sylan9bb 680	1 ⊢ ((A = B ∧ C = D) → (A = C ↔ B = D))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-an 360 df-ex 1542 df-cleq 2346

This theorem is referenced by: eqeq12i 2366 eqeq12d 2367 eqeqan12d 2368 dfpw12 4301 fununiq 5517 fntxp 5804 pw1fnf1o 5855 fundmen 6043 ncdisjeq 6148 peano4nc 6150 sbth 6206 tc11 6228 fnfrec 6320