Theorem rabbii 2746

Description: Equivalent wff's correspond to equal restricted class abstractions. Inference form of rabbidv 2749. (Contributed by Peter Mazsa, 1-Nov-2019.)

Hypothesis

Ref	Expression
rabbii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
rabbii	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}

Proof of Theorem rabbii

Step	Hyp	Ref	Expression
1		rabbii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	a1i 9	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
3	2	rabbiia 2745	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {crab 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-rab 2481

This theorem is referenced by:  dfdif3  3269  suplocexpr  7785  dfrhm2  13650  dmtopon  14191