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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 2806 | . 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}) |
3 | df-rab 3434 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} | |
4 | df-rab 3434 | . 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)} | |
5 | 2, 3, 4 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 {crab 3433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-rab 3434 |
This theorem is referenced by: rabbia2 3436 rabbidva 3440 rabeq 3448 rabeqbidva 3450 extmptsuppeq 8212 dfac2a 10168 hashbclem 14488 umgrislfupgrlem 29154 wwlksn0s 29891 wwlksnextwrd 29927 wpthswwlks2on 29991 rusgrnumwwlkl1 29998 clwwlknon1 30126 orvcgteel 34449 orvclteel 34454 wevgblacfn 35093 mapdvalc 41612 mapdval4N 41615 ovncvrrp 46520 ovnsubaddlem1 46526 ovnsubadd 46528 ovncvr2 46567 hspmbl 46585 smflim 46733 smflimsuplem1 46776 smflimsuplem3 46778 smflimsuplem7 46782 smflimsup 46784 |
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