Theorem bj-rabeqd 32916

Description: Deduction form of rabeq 3192. Note that contrary to rabeq 3192 it has no dv condition. (Contributed by BJ, 27-Apr-2019.)

Hypotheses

Ref	Expression
bj-rabeqd.nf	⊢ Ⅎ𝑥𝜑
bj-rabeqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

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

Proof of Theorem bj-rabeqd

Step	Hyp	Ref	Expression
1		bj-rabeqd.nf	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-rabeqd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eleq2 2690	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
4	3	anbi1d 741	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))
5	2, 4	syl 17	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))
6	1, 5	bj-rabbida2 32913	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921

This theorem is referenced by:  bj-rabeqbid  32917  bj-rabeqbida  32918  bj-inrab2  32924