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Mirrors > Home > MPE Home > Th. List > rabbidva2 | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
Ref | Expression |
---|---|
rabbidva2.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
Ref | Expression |
---|---|
rabbidva2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbidva2.1 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) | |
2 | 1 | abbidv 2807 | . 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}) |
3 | df-rab 3073 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
4 | df-rab 3073 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)} | |
5 | 2, 3, 4 | 3eqtr4g 2803 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {cab 2715 {crab 3068 |
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 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-rab 3073 |
This theorem is referenced by: rabbia2 3412 rabbidva 3413 rabeq 3418 extmptsuppeq 8004 dfac2a 9885 hashbclem 14164 umgrislfupgrlem 27492 wwlksn0s 28226 wwlksnextwrd 28262 wpthswwlks2on 28326 rusgrnumwwlkl1 28333 clwwlknon1 28461 orvcgteel 32434 orvclteel 32439 mapdvalc 39643 mapdval4N 39646 ovncvrrp 44102 ovnsubaddlem1 44108 ovnsubadd 44110 ovncvr2 44149 hspmbl 44167 smflim 44312 smflimsuplem1 44353 smflimsuplem3 44355 smflimsuplem7 44359 smflimsup 44361 |
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