Theorem 3sstr3d 4012

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Hypotheses

Ref	Expression
3sstr3d.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3sstr3d.2	⊢ (𝜑 → 𝐴 = 𝐶)
3sstr3d.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
3sstr3d	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Proof of Theorem 3sstr3d

Step	Hyp	Ref	Expression
1		3sstr3d.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		3sstr3d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
3		3sstr3d.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 3	sseq12d 3999	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷))
5	1, 4	mpbid 234	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1533   ⊆ wss 3935

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-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951

This theorem is referenced by:  cnvtsr  17826  dprdss  19145  dprd2da  19158  dmdprdsplit2lem  19161  mplind  20276  txcmplem1  22243  setsmstopn  23082  tngtopn  23253  bcthlem2  23922  bcthlem4  23924  uniiccvol  24175  dyadmaxlem  24192  dvlip2  24586  dvne0  24602  shlej2  29132  gsumzresunsn  30686  pmtrcnel2  30729  cyc3co2  30777  fedgmullem1  31020  hauseqcn  31133  bnd2lem  35063  heiborlem8  35090  dochord  38500  lclkrlem2p  38652  mapdsn  38771  hbtlem5  39721  fvmptiunrelexplb0d  40022  fvmptiunrelexplb1d  40024