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Theorem ss2abdvALT 3995
Description: Alternate proof of ss2abdv 3994. Shorter, but requiring ax-8 2114. (Contributed by Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ss2abdvALT.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
ss2abdvALT (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem ss2abdvALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ss2abdvALT.1 . . . 4 (𝜑 → (𝜓𝜒))
21sbimdv 2083 . . 3 (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
3 df-clab 2780 . . 3 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
4 df-clab 2780 . . 3 (𝑦 ∈ {𝑥𝜒} ↔ [𝑦 / 𝑥]𝜒)
52, 3, 43imtr4g 299 . 2 (𝜑 → (𝑦 ∈ {𝑥𝜓} → 𝑦 ∈ {𝑥𝜒}))
65ssrdv 3924 1 (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2069  wcel 2112  {cab 2779  wss 3884
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 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901
This theorem is referenced by: (None)
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