Theorem eqssd 2962

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 2960	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
4	1, 2, 3	sylanbrc 394	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  eqrd  2963  unissel  3609  intmin  3635  int0el  3645  dmcosseq  4603  relfld  4846  imadif  4979  imain  4981  fimacnv  5296  fo2ndf  5848  tposeq  5862  tfrlemibfn  5942  tfrlemi14d  5947  nndifsnid  6080  fidifsnid  6332  fisbth  6340  addnqpr  6659  mulnqpr  6675  distrprg  6686  ltexpri  6711  addcanprg  6714  recexprlemex  6735  aptipr  6739  cauappcvgprlemladd  6756  fzopth  8924  fzosplit  9033  fzouzsplit  9035  frecuzrdgfn  9198  findset  10070