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Theorem rabbidva2 3174
 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 2738 . 2 (𝜑 → {𝑥 ∣ (𝑥𝐴𝜓)} = {𝑥 ∣ (𝑥𝐵𝜒)})
3 df-rab 2916 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
4 df-rab 2916 . 2 {𝑥𝐵𝜒} = {𝑥 ∣ (𝑥𝐵𝜒)}
52, 3, 43eqtr4g 2680 1 (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐵𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  {cab 2607  {crab 2911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-rab 2916 This theorem is referenced by:  rabbia2  3175  extmptsuppeq  7264  dfac2a  8896  hashbclem  13174  umgrislfupgrlem  25912  wwlksn0s  26615  wwlksnextwrd  26661  wpthswwlks2on  26722  rusgrnumwwlkl1  26730  numclwwlkovf2  27073  orvcgteel  30310  orvclteel  30315  mapdvalc  36398  mapdval4N  36401  ovncvrrp  40085  ovnsubaddlem1  40091  ovnsubadd  40093  ovncvr2  40132  hspmbl  40150  smflim  40292  smflimsuplem1  40333  smflimsuplem3  40335  smflimsuplem7  40339  smflimsup  40341
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