Theorem ssrd 3940

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 2221	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		ssrd.1	. . 3 ⊢ Ⅎ𝑥𝐴
5		ssrd.2	. . 3 ⊢ Ⅎ𝑥𝐵
6	4, 5	dfssf 3926	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	3, 6	sylibr 234	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884   ⊆ wss 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-11 2163  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-nf 1786  df-clel 2812  df-nfc 2886  df-ss 3920

This theorem is referenced by:  rabss3d  4035  neiptopnei  23088  topdifinffinlem  37602  relowlssretop  37618  ralssiun  37662  sticksstones1  42516  sticksstones11  42526  ssdf2  45500  ssfiunibd  45671  stoweidlem52  46410  stoweidlem59  46417