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Theorem rabbidva2 3386
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 2807 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
3 df-rab 3070 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
4 df-rab 3070 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
52, 3, 43eqtr4g 2803 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  {cab 2714  {crab 3065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-rab 3070
This theorem is referenced by:  rabbia2  3387  rabbidva  3388  rabeq  3394  extmptsuppeq  7930  dfac2a  9743  hashbclem  14016  umgrislfupgrlem  27213  wwlksn0s  27945  wwlksnextwrd  27981  wpthswwlks2on  28045  rusgrnumwwlkl1  28052  clwwlknon1  28180  orvcgteel  32146  orvclteel  32151  mapdvalc  39380  mapdval4N  39383  ovncvrrp  43777  ovnsubaddlem1  43783  ovnsubadd  43785  ovncvr2  43824  hspmbl  43842  smflim  43984  smflimsuplem1  44025  smflimsuplem3  44027  smflimsuplem7  44031  smflimsup  44033
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