Theorem eleq12d 2211

Description: Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Hypotheses

Ref	Expression
eleq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
eleq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
eleq12d	⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))

Proof of Theorem eleq12d

Step	Hyp	Ref	Expression
1		eleq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
2	1	eleq2d 2210	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷))
3		eleq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	eleq1d 2209	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷))
5	2, 4	bitrd 187	1 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ∈ wcel 1481

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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136

This theorem is referenced by:  cbvraldva2  2664  cbvrexdva2  2665  cdeqel  2909  ru  2912  sbcel12g  3022  cbvralcsf  3067  cbvrexcsf  3068  cbvreucsf  3069  cbvrabcsf  3070  onintexmid  4495  elvvuni  4611  elrnmpt1  4798  smoeq  6195  smores  6197  smores2  6199  iordsmo  6202  nnaordi  6412  nnaordr  6414  fvixp  6605  cbvixp  6617  mptelixpg  6636  exmidaclem  7081  cc1  7097  cc2lem  7098  cc3  7100  ltapig  7170  ltmpig  7171  fzsubel  9871  elfzp1b  9908  ennnfonelemg  11952  ennnfonelemp1  11955  ennnfonelemnn0  11971  ctiunctlemu1st  11983  ctiunctlemu2nd  11984  ctiunctlemudc  11986  ctiunctlemfo  11988  istps  12238  tpspropd  12242  eltpsg  12246  isms  12661  mspropd  12686  cnlimci  12850