Theorem rabbia2 3418

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 3417	. 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
4	3	mptru 1547	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2108  {crab 3415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-rab 3416

This theorem is referenced by:  rabbiia  3419  rabswap  3425  rabeqi  3429  rabrabi  3435  f1ossf1o  7118  finsumvtxdg2ssteplem3  29527  clwlknf1oclwwlkn  30065  clwwlknon2x  30084  numclwwlkovh  30354  ballotlem2  34521  smflim  46806  smflim2  46835  smflimsuplem1  46849  smflimsup  46857  sprvalpwn0  47497