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Theorem rabbidva2 3217
 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 2770 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
3 df-rab 2950 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
4 df-rab 2950 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
52, 3, 43eqtr4g 2710 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  {crab 2945 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-rab 2950 This theorem is referenced by:  rabbia2  3218  extmptsuppeq  7364  dfac2a  8990  hashbclem  13274  umgrislfupgrlem  26062  wwlksn0s  26815  wwlksnextwrd  26860  wpthswwlks2on  26927  wpthswwlks2onOLD  26928  rusgrnumwwlkl1  26935  clwwlknon1  27072  orvcgteel  30657  orvclteel  30662  mapdvalc  37235  mapdval4N  37238  ovncvrrp  41099  ovnsubaddlem1  41105  ovnsubadd  41107  ovncvr2  41146  hspmbl  41164  smflim  41306  smflimsuplem1  41347  smflimsuplem3  41349  smflimsuplem7  41353  smflimsup  41355
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