Theorem rabbida 3430

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3410 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2165, ax-11 2181. (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 586	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	1, 3	rabbida4 3429	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550  Ⅎwnf 1793   ∈ wcel 2132  {crab 3404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-9 2142  ax-12 2202  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-rab 3405

This theorem is referenced by:  rabbid  3431  rabeqbida  3433  smfpimltmpt  47258  smfpimltxrmptf  47270  smfpimgtmpt  47293  smfpimgtxrmptf  47296  smfrec  47301  smfsupmpt  47327  smfinflem  47329  smfinfmpt  47331