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Theorem clss2lem 40729
 Description: The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.)
Hypothesis
Ref Expression
clss2lem.1 (𝜑 → (𝜒𝜓))
Assertion
Ref Expression
clss2lem (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} ⊆ {𝑥 ∣ (𝑋𝑥𝜒)})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝑋(𝑥)

Proof of Theorem clss2lem
StepHypRef Expression
1 clss2lem.1 . . . . 5 (𝜑 → (𝜒𝜓))
21adantld 494 . . . 4 (𝜑 → ((𝑋𝑥𝜒) → 𝜓))
32alrimiv 1928 . . 3 (𝜑 → ∀𝑥((𝑋𝑥𝜒) → 𝜓))
4 pm5.3 576 . . . . 5 (((𝑋𝑥𝜒) → 𝜓) ↔ ((𝑋𝑥𝜒) → (𝑋𝑥𝜓)))
54albii 1821 . . . 4 (∀𝑥((𝑋𝑥𝜒) → 𝜓) ↔ ∀𝑥((𝑋𝑥𝜒) → (𝑋𝑥𝜓)))
6 ss2ab 3966 . . . 4 ({𝑥 ∣ (𝑋𝑥𝜒)} ⊆ {𝑥 ∣ (𝑋𝑥𝜓)} ↔ ∀𝑥((𝑋𝑥𝜒) → (𝑋𝑥𝜓)))
75, 6bitr4i 281 . . 3 (∀𝑥((𝑋𝑥𝜒) → 𝜓) ↔ {𝑥 ∣ (𝑋𝑥𝜒)} ⊆ {𝑥 ∣ (𝑋𝑥𝜓)})
83, 7sylib 221 . 2 (𝜑 → {𝑥 ∣ (𝑋𝑥𝜒)} ⊆ {𝑥 ∣ (𝑋𝑥𝜓)})
9 intss 4862 . 2 ({𝑥 ∣ (𝑋𝑥𝜒)} ⊆ {𝑥 ∣ (𝑋𝑥𝜓)} → {𝑥 ∣ (𝑋𝑥𝜓)} ⊆ {𝑥 ∣ (𝑋𝑥𝜒)})
108, 9syl 17 1 (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} ⊆ {𝑥 ∣ (𝑋𝑥𝜒)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  {cab 2735   ⊆ wss 3860  ∩ cint 4841 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-v 3411  df-in 3867  df-ss 3877  df-int 4842 This theorem is referenced by: (None)
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