Theorem rabeq12f 38524

Description: Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018.)

Hypotheses

Ref	Expression
rabeq12f.1	⊢ Ⅎ𝑥𝐴
rabeq12f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
rabeq12f	⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓})

Proof of Theorem rabeq12f

Step	Hyp	Ref	Expression
1		rabbi 3421	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓})
2	1	biimpi 217	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓})
3		rabeq12f.1	. . 3 ⊢ Ⅎ𝑥𝐴
4		rabeq12f.2	. . 3 ⊢ Ⅎ𝑥𝐵
5	3, 4	rabeqf 3425	. 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
6	2, 5	sylan9eqr 2796	1 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)) → {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  Ⅎwnfc 2886  ∀wral 3053  {crab 3391

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rab 3392

This theorem is referenced by: (None)