Theorem sseqtrid 3205

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 3179	. . 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 1353   ⊆ wss 3129

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 3135  df-ss 3142

This theorem is referenced by:  fssdm  5380  fndmdif  5621  fneqeql2  5625  fconst4m  5736  f1opw2  6076  ecss  6575  fopwdom  6835  ssenen  6850  phplem2  6852  fiintim  6927  casefun  7083  caseinj  7087  djufun  7102  djuinj  7104  nn0supp  9227  monoord2  10476  binom1dif  11494  cnpnei  13689  cnntri  13694  cnntr  13695  cncnp  13700  cndis  13711  txdis1cn  13748  hmeontr  13783  hmeoimaf1o  13784  dvcoapbr  14141