Theorem sylan9req 2134

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 2086	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		sylan9req.2	. 2 ⊢ (𝜓 → 𝐵 = 𝐶)
4	2, 3	sylan9eq 2133	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074

This theorem is referenced by:  xpid11m  4575  fndmu  5020  fodmrnu  5134  funcoeqres  5177  fvunsng  5378  prarloclem5  6690  addlocprlemeq  6723  zdiv  8435  resqrexlemnm  9904  dvdsmulc  10223  cncongrcoprm  10488