Theorem bj-gabeqd 35104

Description: Equality of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)

Hypotheses

Ref	Expression
bj-gabeqd.nf	⊢ (𝜑 → ∀𝑥𝜑)
bj-gabeqd.c	⊢ (𝜑 → 𝐴 = 𝐵)
bj-gabeqd.f	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem bj-gabeqd

Step	Hyp	Ref	Expression
1		bj-gabeqd.nf	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-gabeqd.c	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3		bj-gabeqd.f	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	3	biimpd 228	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	1, 2, 4	bj-gabssd 35103	. 2 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})
6	2	eqcomd 2745	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
7	3	biimprd 247	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
8	1, 6, 7	bj-gabssd 35103	. 2 ⊢ (𝜑 → {𝐵 ∣ 𝑥 ∣ 𝜒} ⊆ {𝐴 ∣ 𝑥 ∣ 𝜓})
9	5, 8	eqssd 3942	1 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541  {bj-cgab 35100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-v 3432  df-in 3898  df-ss 3908  df-bj-gab 35101

This theorem is referenced by: (None)