Theorem rabbida 3442

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3422 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2177, ax-11 2193. (Revised by Wolf Lammen, 14-Mar-2025.)

Hypotheses

Ref	Expression
rabbida.n	⊢ Ⅎ𝑥𝜑
rabbida.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rabbida	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Proof of Theorem rabbida

Step	Hyp	Ref	Expression
1		rabbida.n	. 2 ⊢ Ⅎ𝑥𝜑
2		rabbida.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	pm5.32da 587	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	1, 3	rabbida4 3441	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  Ⅎwnf 1805   ∈ wcel 2144  {crab 3416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-9 2154  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-rab 3417

This theorem is referenced by:  rabbid  3443  rabeqbida  3445  smfpimltmpt  47325  smfpimltxrmptf  47337  smfpimgtmpt  47360  smfpimgtxrmptf  47363  smfrec  47368  smfsupmpt  47394  smfinflem  47396  smfinfmpt  47398