Theorem bj-gabeqd 36903

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 229	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	1, 2, 4	bj-gabssd 36902	. 2 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})
6	2	eqcomd 2746	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
7	3	biimprd 248	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
8	1, 6, 7	bj-gabssd 36902	. 2 ⊢ (𝜑 → {𝐵 ∣ 𝑥 ∣ 𝜒} ⊆ {𝐴 ∣ 𝑥 ∣ 𝜓})
9	5, 8	eqssd 4026	1 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537  {bj-cgab 36899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ss 3993  df-bj-gab 36900

This theorem is referenced by: (None)