Theorem sseqtrid 3290

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 3264	. . 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 1398   ⊆ wss 3213

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3219  df-ss 3226

This theorem is referenced by:  fssdm  5526  fndmdif  5785  fneqeql2  5789  fconst4m  5906  f1opw2  6263  fsuppeq  6449  fsuppeqg  6450  ecss  6812  pw2f1odclem  7089  fopwdom  7091  ssenen  7107  phplem2  7109  fiintim  7193  casefun  7378  caseinj  7382  djufun  7397  djuinj  7399  nn0supp  9557  monoord2  10855  binom1dif  12181  znleval  14850  cnpnei  15133  cnntri  15138  cnntr  15139  cncnp  15144  cndis  15155  txdis1cn  15192  hmeontr  15227  hmeoimaf1o  15228  dvcoapbr  15621  uhgrspansubgr  16321  vtxdfifiun  16341