Theorem bj-gabssd 35181

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 512	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ (𝜓 → 𝜒)))
5	1, 4	alrimih 1825	. 2 ⊢ (𝜑 → ∀𝑥(𝐴 = 𝐵 ∧ (𝜓 → 𝜒)))
6		bj-gabss 35180	. 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 1538   = wceq 1540   ⊆ wss 3897  {bj-cgab 35178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-v 3443  df-in 3904  df-ss 3914  df-bj-gab 35179

This theorem is referenced by:  bj-gabeqd  35182