Theorem eqneltrd 2932

Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
eqneltrd.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqneltrd.2	⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
eqneltrd	⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)

Proof of Theorem eqneltrd

Step	Hyp	Ref	Expression
1		eqneltrd.2	. 2 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)
2		eqneltrd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	eleq1d 2897	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
4	1, 3	mtbird 327	1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-cleq 2814  df-clel 2893

This theorem is referenced by:  eqneltrrd  2933  opabn1stprc  7750  omopth2  8204  fpwwe2  10059  znnn0nn  12088  sqrtneglem  14620  dvdsaddre2b  15651  2mulprm  16031  mreexmrid  16908  mplcoe1  20240  mplcoe5  20243  2sqn0  26004  nn0xmulclb  30490  pmtrcnel  30728  cycpmco2lem5  30767  extdg1id  31048  reprpmtf1o  31892  fmlafvel  32627  fvnobday  33178  bj-snmooreb  34400  islln2a  36647  islpln2a  36678  islvol2aN  36722  oddfl  41536  sumnnodd  41904  sinaover2ne0  42142  dvnprodlem1  42224  dirker2re  42371  dirkerdenne0  42372  dirkertrigeqlem3  42379  dirkercncflem1  42382  dirkercncflem2  42383  dirkercncflem4  42385  fouriersw  42510  sqrtnegnre  43501  requad01  43780