Theorem sylan9eqr 2407

Description: An equality transitivity deduction. (Contributed by NM, 8-May-1994.)

Hypotheses

Ref	Expression
sylan9eqr.1	⊢ (φ → A = B)
sylan9eqr.2	⊢ (ψ → B = C)

Assertion

Ref	Expression
sylan9eqr	⊢ ((ψ ∧ φ) → A = C)

Proof of Theorem sylan9eqr

Step	Hyp	Ref	Expression
1		sylan9eqr.1	. . 3 ⊢ (φ → A = B)
2		sylan9eqr.2	. . 3 ⊢ (ψ → B = C)
3	1, 2	sylan9eq 2405	. 2 ⊢ ((φ ∧ ψ) → A = C)
4	3	ancoms 439	1 ⊢ ((ψ ∧ φ) → 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:  sbcied2  3084  csbied2  3180  fun2ssres  5146  funssfv  5344  caovmo  5646  fmpt2x  5731  freceq12  6312