Theorem ssrd 2950

Description: Deduction rule 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 1415	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		ssrd.1	. . 3 ⊢ Ⅎ𝑥𝐴
5		ssrd.2	. . 3 ⊢ Ⅎ𝑥𝐵
6	4, 5	dfss2f 2936	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	3, 6	sylibr 137	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349   ∈ wcel 1393  Ⅎwnfc 2165   ⊆ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931

This theorem is referenced by:  eqrd  2963