Theorem sylan9req 2406

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 2358	. 2 ⊢ (φ → A = B)
3		sylan9req.2	. 2 ⊢ (ψ → B = C)
4	2, 3	sylan9eq 2405	1 ⊢ ((φ ∧ ψ) → A = C)

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

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  fndmu  5185  fodmrnu  5278