Theorem bj-gabssd 37494

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	⊢ (𝜑 → 𝐴 = 𝐵)
bj-gabssd.f	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
bj-gabssd	⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})

Proof of Theorem bj-gabssd

Step	Hyp	Ref	Expression
1		bj-gabssd.nf	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-gabssd.c	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
3		bj-gabssd.f	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
4	2, 3	jca 520	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ (𝜓 → 𝜒)))
5	1, 4	alrimih 1851	. 2 ⊢ (𝜑 → ∀𝑥(𝐴 = 𝐵 ∧ (𝜓 → 𝜒)))
6		bj-gabss 37493	. 2 ⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜓 → 𝜒)) → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})
7	5, 6	syl 18	1 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565   = wceq 1567   ⊆ wss 3913  {bj-cgab 37491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ss 3930  df-bj-gab 37492

This theorem is referenced by:  bj-gabeqd  37495