Theorem eqssd 3016

Description: Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)

Hypotheses

Ref	Expression
eqssd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
eqssd.2	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
eqssd	⊢ (𝜑 → 𝐴 = 𝐵)

Proof of Theorem eqssd

Step	Hyp	Ref	Expression
1		eqssd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		eqssd.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
3		eqss 3014	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
4	1, 2, 3	sylanbrc 408	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1284   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  eqrd  3017  unissel  3630  intmin  3656  int0el  3666  dmcosseq  4621  relfld  4866  imadif  4999  imain  5001  fimacnv  5317  fo2ndf  5868  tposeq  5885  tfrlemibfn  5965  tfrlemi14d  5970  nndifsnid  6103  fidifsnid  6356  fisbth  6367  en2eqpr  6380  addnqpr  6751  mulnqpr  6767  distrprg  6778  ltexpri  6803  addcanprg  6806  recexprlemex  6827  aptipr  6831  cauappcvgprlemladd  6848  fzopth  9079  fzosplit  9186  fzouzsplit  9188  frecuzrdgfn  9414  findset  10740