Theorem rabbia2 3392

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 3391	. 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
4	3	mptru 1549	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  {crab 3389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-rab 3390

This theorem is referenced by:  rabbiia  3393  rabswap  3398  rabeqi  3402  rabrabi  3408  rabrab  3413  f1ossf1o  7081  finsumvtxdg2ssteplem3  29616  clwlknf1oclwwlkn  30154  clwwlknon2x  30173  numclwwlkovh  30443  ballotlem2  34633  fineqvnttrclse  35268  smflim  47205  smflim2  47234  smflimsuplem1  47248  smflimsup  47256  sprvalpwn0  47943