Theorem rabssd 45152

Description: Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
rabssd.1	⊢ Ⅎ𝑥𝜑
rabssd.2	⊢ Ⅎ𝑥𝐵
rabssd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
rabssd	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵)

Proof of Theorem rabssd

Step	Hyp	Ref	Expression
1		rabssd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rabssd.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝑥 ∈ 𝐵)
3	2	3exp 1119	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝑥 ∈ 𝐵)))
4	1, 3	ralrimi 3256	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝑥 ∈ 𝐵))
5		rabssd.2	. . 3 ⊢ Ⅎ𝑥𝐵
6	5	rabssf 45129	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜒 → 𝑥 ∈ 𝐵))
7	4, 6	sylibr 234	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1086  Ⅎwnf 1782   ∈ wcel 2107  Ⅎwnfc 2889  ∀wral 3060  {crab 3435   ⊆ wss 3950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rab 3436  df-ss 3967

This theorem is referenced by:  pimxrneun  45504  fsupdm  46862  finfdm  46866