Theorem rabbidva2 3404

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Hypothesis

Ref	Expression
rabbidva2.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))

Assertion

Ref	Expression
rabbidva2	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rabbidva2

Step	Hyp	Ref	Expression
1		rabbidva2.1	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
2	1	abbidv 2795	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)})
3		df-rab 3403	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
4		df-rab 3403	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}
5	2, 3, 4	3eqtr4g 2789	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  {crab 3402

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 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-rab 3403

This theorem is referenced by:  rabbia2  3405  rabbidva  3409  rabeq  3417  rabeqbidva  3419  extmptsuppeq  8144  dfac2a  10059  hashbclem  14393  n0cutlt  28225  umgrislfupgrlem  29025  wwlksn0s  29764  wwlksnextwrd  29800  wpthswwlks2on  29864  rusgrnumwwlkl1  29871  clwwlknon1  29999  orvcgteel  34432  orvclteel  34437  wevgblacfn  35069  mapdvalc  41596  mapdval4N  41599  ovncvrrp  46535  ovnsubaddlem1  46541  ovnsubadd  46543  ovncvr2  46582  hspmbl  46600  smflim  46748  smflimsuplem1  46791  smflimsuplem3  46793  smflimsuplem7  46797  smflimsup  46799  initopropd  49205  termopropd  49206