Theorem ssrd 3918

Description: Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Hypotheses

Ref	Expression
ssrd.0	⊢ Ⅎ𝑥𝜑
ssrd.1	⊢ Ⅎ𝑥𝐴
ssrd.2	⊢ Ⅎ𝑥𝐵
ssrd.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))

Assertion

Ref	Expression
ssrd	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Proof of Theorem ssrd

Step	Hyp	Ref	Expression
1		ssrd.0	. . 3 ⊢ Ⅎ𝑥𝜑
2		ssrd.3	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
3	1, 2	alrimi 2253	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		ssrd.1	. . 3 ⊢ Ⅎ𝑥𝐴
5		ssrd.2	. . 3 ⊢ Ⅎ𝑥𝐵
6	4, 5	dfssf 3904	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	3, 6	sylibr 237	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816   ∈ wcel 2147  Ⅎwnfc 2911   ⊆ wss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-clel 2839  df-nfc 2913  df-ss 3901

This theorem is used by:  rabss3d  4009  neiptopnei  23505  onprcf1acwevdlem1  36086  topdifinffinlem  38538  relowlssretop  38554  ralssiun  38598  sticksstones1  43462  sticksstones11  43472  ssdf2  46429  ssfiunibd  46598  stoweidlem52  47335  stoweidlem59  47342