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Theorem ss2ab1 41707
Description: Class abstractions in a subclass relationship, closed form. One direction of ss2ab 4054 using fewer axioms. (Contributed by SN, 22-Dec-2024.)
Assertion
Ref Expression
ss2ab1 (∀𝑥(𝜑𝜓) → {𝑥𝜑} ⊆ {𝑥𝜓})

Proof of Theorem ss2ab1
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 spsbim 2068 . . 3 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
2 df-clab 2706 . . 3 (𝑡 ∈ {𝑥𝜑} ↔ [𝑡 / 𝑥]𝜑)
3 df-clab 2706 . . 3 (𝑡 ∈ {𝑥𝜓} ↔ [𝑡 / 𝑥]𝜓)
41, 2, 33imtr4g 296 . 2 (∀𝑥(𝜑𝜓) → (𝑡 ∈ {𝑥𝜑} → 𝑡 ∈ {𝑥𝜓}))
54ssrdv 3986 1 (∀𝑥(𝜑𝜓) → {𝑥𝜑} ⊆ {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1532  [wsb 2060  wcel 2099  {cab 2705  wss 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964
This theorem is referenced by: (None)
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