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Theorem rabbidva2 3435
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.)
Hypothesis
Ref Expression
rabbidva2.1 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
Assertion
Ref Expression
rabbidva2 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rabbidva2
StepHypRef Expression
1 rabbidva2.1 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))
21abbidv 2802 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
3 df-rab 3434 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
4 df-rab 3434 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
52, 3, 43eqtr4g 2798 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-rab 3434
This theorem is referenced by:  rabbia2  3436  rabbidva  3440  rabeq  3447  extmptsuppeq  8173  dfac2a  10124  hashbclem  14411  umgrislfupgrlem  28382  wwlksn0s  29115  wwlksnextwrd  29151  wpthswwlks2on  29215  rusgrnumwwlkl1  29222  clwwlknon1  29350  orvcgteel  33466  orvclteel  33471  mapdvalc  40500  mapdval4N  40503  ovncvrrp  45280  ovnsubaddlem1  45286  ovnsubadd  45288  ovncvr2  45327  hspmbl  45345  smflim  45493  smflimsuplem1  45536  smflimsuplem3  45538  smflimsuplem7  45542  smflimsup  45544
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