Theorem 3sstr4g 3200

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4g.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3sstr4g.2	⊢ 𝐶 = 𝐴
3sstr4g.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3sstr4g	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Proof of Theorem 3sstr4g

Step	Hyp	Ref	Expression
1		3sstr4g.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		3sstr4g.2	. . 3 ⊢ 𝐶 = 𝐴
3		3sstr4g.3	. . 3 ⊢ 𝐷 = 𝐵
4	2, 3	sseq12i 3185	. 2 ⊢ (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	sylibr 134	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1353   ⊆ wss 3131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3137  df-ss 3144

This theorem is referenced by:  rabss2  3240  unss2  3308  sslin  3363  ssopab2  4277  xpss12  4735  coss1  4784  coss2  4785  cnvss  4802  rnss  4859  ssres  4935  ssres2  4936  imass1  5005  imass2  5006  imadif  5298  imain  5300  ssoprab2  5934  suppssfv  6082  suppssov1  6083  tposss  6250  ss2ixp  6714  isumsplit  11502  isumrpcl  11505