Theorem ssdf 40178

Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
ssdf.1	⊢ Ⅎ𝑥𝜑
ssdf.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)

Assertion

Ref	Expression
ssdf	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssdf

Step	Hyp	Ref	Expression
1		ssdf.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ssdf.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	2	ex 403	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
4	1, 3	ralrimi 3139	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
5		dfss3 3810	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
6	4, 5	sylibr 226	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 386  Ⅎwnf 1827   ∈ wcel 2107  ∀wral 3090   ⊆ wss 3792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ral 3095  df-in 3799  df-ss 3806

This theorem is referenced by:  ssd  40183  smfaddlem2  41899  smfadd  41900  smfmullem4  41928  smfmul  41929  smflimsuplem4  41956