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Theorem bj-gabssd 36919
Description: Inclusion of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)
Hypotheses
Ref Expression
bj-gabssd.nf (𝜑 → ∀𝑥𝜑)
bj-gabssd.c (𝜑𝐴 = 𝐵)
bj-gabssd.f (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
bj-gabssd (𝜑 → {𝐴𝑥𝜓} ⊆ {𝐵𝑥𝜒})

Proof of Theorem bj-gabssd
StepHypRef Expression
1 bj-gabssd.nf . . 3 (𝜑 → ∀𝑥𝜑)
2 bj-gabssd.c . . . 4 (𝜑𝐴 = 𝐵)
3 bj-gabssd.f . . . 4 (𝜑 → (𝜓𝜒))
42, 3jca 511 . . 3 (𝜑 → (𝐴 = 𝐵 ∧ (𝜓𝜒)))
51, 4alrimih 1821 . 2 (𝜑 → ∀𝑥(𝐴 = 𝐵 ∧ (𝜓𝜒)))
6 bj-gabss 36918 . 2 (∀𝑥(𝐴 = 𝐵 ∧ (𝜓𝜒)) → {𝐴𝑥𝜓} ⊆ {𝐵𝑥𝜒})
75, 6syl 17 1 (𝜑 → {𝐴𝑥𝜓} ⊆ {𝐵𝑥𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wss 3963  {bj-cgab 36916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ss 3980  df-bj-gab 36917
This theorem is referenced by:  bj-gabeqd  36920
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