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Theorem clss2lem 37385
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 483 . . . 4 (𝜑 → ((𝑋𝑥𝜒) → 𝜓))
32alrimiv 1857 . . 3 (𝜑 → ∀𝑥((𝑋𝑥𝜒) → 𝜓))
4 pm5.3 747 . . . . 5 (((𝑋𝑥𝜒) → 𝜓) ↔ ((𝑋𝑥𝜒) → (𝑋𝑥𝜓)))
54albii 1744 . . . 4 (∀𝑥((𝑋𝑥𝜒) → 𝜓) ↔ ∀𝑥((𝑋𝑥𝜒) → (𝑋𝑥𝜓)))
6 ss2ab 3654 . . . 4 ({𝑥 ∣ (𝑋𝑥𝜒)} ⊆ {𝑥 ∣ (𝑋𝑥𝜓)} ↔ ∀𝑥((𝑋𝑥𝜒) → (𝑋𝑥𝜓)))
75, 6bitr4i 267 . . 3 (∀𝑥((𝑋𝑥𝜒) → 𝜓) ↔ {𝑥 ∣ (𝑋𝑥𝜒)} ⊆ {𝑥 ∣ (𝑋𝑥𝜓)})
83, 7sylib 208 . 2 (𝜑 → {𝑥 ∣ (𝑋𝑥𝜒)} ⊆ {𝑥 ∣ (𝑋𝑥𝜓)})
9 intss 4468 . 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 384  wal 1478  {cab 2612  wss 3560   cint 4445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-in 3567  df-ss 3574  df-int 4446
This theorem is referenced by: (None)
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