Theorem ssrd 3152

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 1515	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4		ssrd.1	. . 3 ⊢ Ⅎ𝑥𝐴
5		ssrd.2	. . 3 ⊢ Ⅎ𝑥𝐵
6	4, 5	dfss2f 3138	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
7	3, 6	sylibr 133	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4  ∀wal 1346  Ⅎwnf 1453   ∈ wcel 2141  Ⅎwnfc 2299   ⊆ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134

This theorem is referenced by:  eqrd  3165  exmidomni  7118