Theorem rabbida 3375

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3379 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)

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	. . 3 ⊢ Ⅎ𝑥𝜑
2		rabbida.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	ex 416	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
4	1, 3	ralrimi 3128	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒))
5		rabbi 3286	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
6	4, 5	sylib 221	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542  Ⅎwnf 1790   ∈ wcel 2114  ∀wral 3053  {crab 3057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-ral 3058  df-rab 3062

This theorem is referenced by:  rabbid  3376  bj-rabeqbida  34741  pimgtmnf  43798  smfpimltmpt  43821  smfpimltxrmpt  43833  smfpimgtmpt  43855  smfpimgtxrmpt  43858  smfrec  43862  smfsupmpt  43887  smfinflem  43889  smfinfmpt  43891