Theorem syl5sseq 3047

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

Hypotheses

Ref	Expression
syl5sseq.1	⊢ 𝐵 ⊆ 𝐴
syl5sseq.2	⊢ (𝜑 → 𝐴 = 𝐶)

Assertion

Ref	Expression
syl5sseq	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Proof of Theorem syl5sseq

Step	Hyp	Ref	Expression
1		syl5sseq.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
2		syl5sseq.1	. 2 ⊢ 𝐵 ⊆ 𝐴
3		sseq2 3021	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶))
4	3	biimpa 290	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐶)
5	1, 2, 4	sylancl 404	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1284   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  fndmdif  5293  fneqeql2  5297  fconst4m  5402  f1opw2  5726  ecss  6170  fopwdom  6333  phplem2  6339  nn0supp  8340  monoord2  9456