Theorem bj-gabssd 37298

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 516	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ (𝜓 → 𝜒)))
5	1, 4	alrimih 1831	. 2 ⊢ (𝜑 → ∀𝑥(𝐴 = 𝐵 ∧ (𝜓 → 𝜒)))
6		bj-gabss 37297	. 2 ⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜓 → 𝜒)) → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})
7	5, 6	syl 17	1 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547   ⊆ wss 3883  {bj-cgab 37295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ss 3900  df-bj-gab 37296

This theorem is referenced by:  bj-gabeqd  37299