Theorem sylan9req 2250

Description: An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)

Hypotheses

Ref	Expression
sylan9req.1	⊢ (𝜑 → 𝐵 = 𝐴)
sylan9req.2	⊢ (𝜓 → 𝐵 = 𝐶)

Assertion

Ref	Expression
sylan9req	⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)

Proof of Theorem sylan9req

Step	Hyp	Ref	Expression
1		sylan9req.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2202	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		sylan9req.2	. 2 ⊢ (𝜓 → 𝐵 = 𝐶)
4	2, 3	sylan9eq 2249	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  fndmu  5359  fodmrnu  5488  funcoeqres  5535  fvunsng  5756  prarloclem5  7567  addlocprlemeq  7600  zdiv  9414  resqrexlemnm  11183  fprodssdc  11755  dvdsmulc  11984  cncongrcoprm  12274  mgmidmo  13015  lgsmodeq  15286