Theorem rabbidva2 3414

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 2826	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)})
3		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
4		df-rab 3413	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}
5	2, 3, 4	3eqtr4g 2820	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  {crab 3412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-rab 3413

This theorem is used by:  rabbia2  3415  rabbidva  3418  rabeq  3426  rabeqbidva  3428  rabsneq  4603  extmptsuppeq  8187  dfac2a  10135  hashbclem  14520  n0cutlt  28627  umgrislfupgrlem  29582  wwlksn0s  30332  wwlksnextwrd  30368  wpthswwlks2on  30435  rusgrnumwwlkl1  30442  clwwlknon1  30570  orvcgteel  34982  orvclteel  34987  wevgblacfn  35711  mapdvalc  42505  mapdval4N  42508  ovncvrrp  47395  ovnsubaddlem1  47401  ovnsubadd  47403  ovncvr2  47442  hspmbl  47460  smflim  47608  smflimsuplem1  47651  smflimsuplem3  47653  smflimsuplem7  47657  smflimsup  47659  initopropd  50172  termopropd  50173