Theorem rabbida 3475

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3479 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 415	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝜒)))
4	1, 3	ralrimi 3216	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒))
5		rabbi 3384	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 ↔ 𝜒) ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
6	4, 5	sylib 220	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  ∀wral 3138  {crab 3142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-ral 3143  df-rab 3147

This theorem is referenced by:  rabbid  3476  bj-rabeqbida  34235  pimgtmnf  42993  smfpimltmpt  43016  smfpimltxrmpt  43028  smfpimgtmpt  43050  smfpimgtxrmpt  43053  smfrec  43057  smfsupmpt  43082  smfinflem  43084  smfinfmpt  43086