Theorem sseqtrid 3298

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
sseqtrid.1	⊢ 𝐵 ⊆ 𝐴
sseqtrid.2	⊢ (𝜑 → 𝐴 = 𝐶)

Assertion

Ref	Expression
sseqtrid	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Proof of Theorem sseqtrid

Step	Hyp	Ref	Expression
1		sseqtrid.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
2		sseqtrid.1	. 2 ⊢ 𝐵 ⊆ 𝐴
3		sseq2 3272	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶))
4	3	biimpa 296	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐶)
5	1, 2, 4	sylancl 417	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1402   ⊆ wss 3220

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  fssdm  5547  fndmdif  5808  fneqeql2  5812  fconst4m  5929  f1opw2  6290  fsuppeq  6481  fsuppeqg  6482  ecss  6844  pw2f1odclem  7128  fopwdom  7130  ssenen  7146  phplem2  7148  fiintim  7232  casefun  7419  caseinj  7423  djufun  7438  djuinj  7440  nn0supp  9602  monoord2  10906  binom1dif  12237  ballotfilemro  13249  znleval  14971  cnpnei  15303  cnntri  15308  cnntr  15309  cncnp  15314  cndis  15325  txdis1cn  15362  hmeontr  15397  hmeoimaf1o  15398  dvcoapbr  15791  uhgrspansubgr  16501  vtxdfifiun  16521