Theorem rabbia2 3398

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
rabbia2.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))

Assertion

Ref	Expression
rabbia2	⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}

Proof of Theorem rabbia2

Step	Hyp	Ref	Expression
1		rabbia2.1	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
2	1	a1i 11	. . 3 ⊢ (⊤ → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
3	2	rabbidva2 3397	. 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
4	3	mptru 1548	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ⊤wtru 1542   ∈ wcel 2111  {crab 3395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-rab 3396

This theorem is referenced by:  rabbiia  3399  rabswap  3404  rabeqi  3408  rabrabi  3414  f1ossf1o  7061  finsumvtxdg2ssteplem3  29526  clwlknf1oclwwlkn  30064  clwwlknon2x  30083  numclwwlkovh  30353  ballotlem2  34502  fineqvnttrclse  35144  smflim  46823  smflim2  46852  smflimsuplem1  46866  smflimsup  46874  sprvalpwn0  47522