Theorem sseqtrid 3229

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 3203	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶))
4	3	biimpa 296	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐶)
5	1, 2, 4	sylancl 413	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1364   ⊆ wss 3153

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  fssdm  5418  fndmdif  5663  fneqeql2  5667  fconst4m  5778  f1opw2  6124  ecss  6630  pw2f1odclem  6890  fopwdom  6892  ssenen  6907  phplem2  6909  fiintim  6985  casefun  7144  caseinj  7148  djufun  7163  djuinj  7165  nn0supp  9292  monoord2  10557  binom1dif  11630  znleval  14141  cnpnei  14387  cnntri  14392  cnntr  14393  cncnp  14398  cndis  14409  txdis1cn  14446  hmeontr  14481  hmeoimaf1o  14482  dvcoapbr  14856