Theorem rabbida 3440

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3420 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2174, ax-11 2190. (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 589	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
4	1, 3	rabbida4 3439	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568  Ⅎwnf 1811   ∈ wcel 2141  {crab 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2151  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-rab 3415

This theorem is referenced by:  rabbid  3441  rabeqbida  3443  smfpimltmpt  47408  smfpimltxrmptf  47420  smfpimgtmpt  47443  smfpimgtxrmptf  47446  smfrec  47451  smfsupmpt  47477  smfinflem  47479  smfinfmpt  47481