Theorem difeq2d 3235

Description: Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Hypothesis

Ref	Expression
difeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
difeq2d	⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))

Proof of Theorem difeq2d

Step	Hyp	Ref	Expression
1		difeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		difeq2 3229	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))

Syntax hints:   → wi 4   = wceq 1342   ∖ cdif 3108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-ral 2447  df-rab 2451  df-dif 3113

This theorem is referenced by:  difeq12d  3236  phplem3  6811  phplem4  6812  phplem3g  6813  phplem4dom  6819  phplem4on  6824  fidifsnen  6827  xpfi  6886  sbthlem2  6914  sbthlemi3  6915  isbth  6923  ismkvnex  7110  setsvalg  12361  setsvala  12362  exmid1stab  13714