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Theorem cbvrabcsf 3902
Description: A more general version of cbvrab 3467 with no distinct variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2391. (Contributed by Andrew Salmon, 13-Jul-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvralcsf.1 𝑦𝐴
cbvralcsf.2 𝑥𝐵
cbvralcsf.3 𝑦𝜑
cbvralcsf.4 𝑥𝜓
cbvralcsf.5 (𝑥 = 𝑦𝐴 = 𝐵)
cbvralcsf.6 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrabcsf {𝑥𝐴𝜑} = {𝑦𝐵𝜓}

Proof of Theorem cbvrabcsf
Dummy variables 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1916 . . . 4 𝑧(𝑥𝐴𝜑)
2 nfcsb1v 3881 . . . . . 6 𝑥𝑧 / 𝑥𝐴
32nfcri 2968 . . . . 5 𝑥 𝑧𝑧 / 𝑥𝐴
4 nfs1v 2161 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfan 1901 . . . 4 𝑥(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)
6 id 22 . . . . . 6 (𝑥 = 𝑧𝑥 = 𝑧)
7 csbeq1a 3871 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
86, 7eleq12d 2906 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝑧 / 𝑥𝐴))
9 sbequ12 2254 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
108, 9anbi12d 633 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)))
111, 5, 10cbvab 2891 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑧 ∣ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)}
12 nfcv 2974 . . . . . . 7 𝑦𝑧
13 cbvralcsf.1 . . . . . . 7 𝑦𝐴
1412, 13nfcsb 3884 . . . . . 6 𝑦𝑧 / 𝑥𝐴
1514nfcri 2968 . . . . 5 𝑦 𝑧𝑧 / 𝑥𝐴
16 cbvralcsf.3 . . . . . 6 𝑦𝜑
1716nfsb 2566 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1815, 17nfan 1901 . . . 4 𝑦(𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)
19 nfv 1916 . . . 4 𝑧(𝑦𝐵𝜓)
20 id 22 . . . . . 6 (𝑧 = 𝑦𝑧 = 𝑦)
21 csbeq1 3860 . . . . . . 7 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝑦 / 𝑥𝐴)
22 df-csb 3858 . . . . . . . 8 𝑦 / 𝑥𝐴 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
23 cbvralcsf.2 . . . . . . . . . . . 12 𝑥𝐵
2423nfcri 2968 . . . . . . . . . . 11 𝑥 𝑣𝐵
25 cbvralcsf.5 . . . . . . . . . . . 12 (𝑥 = 𝑦𝐴 = 𝐵)
2625eleq2d 2897 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑣𝐴𝑣𝐵))
2724, 26sbie 2545 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴𝑣𝐵)
28 sbsbc 3753 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑣𝐴[𝑦 / 𝑥]𝑣𝐴)
2927, 28bitr3i 280 . . . . . . . . 9 (𝑣𝐵[𝑦 / 𝑥]𝑣𝐴)
3029abbi2i 2951 . . . . . . . 8 𝐵 = {𝑣[𝑦 / 𝑥]𝑣𝐴}
3122, 30eqtr4i 2847 . . . . . . 7 𝑦 / 𝑥𝐴 = 𝐵
3221, 31syl6eq 2872 . . . . . 6 (𝑧 = 𝑦𝑧 / 𝑥𝐴 = 𝐵)
3320, 32eleq12d 2906 . . . . 5 (𝑧 = 𝑦 → (𝑧𝑧 / 𝑥𝐴𝑦𝐵))
34 sbequ 2091 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
35 cbvralcsf.4 . . . . . . 7 𝑥𝜓
36 cbvralcsf.6 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
3735, 36sbie 2545 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
3834, 37syl6bb 290 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
3933, 38anbi12d 633 . . . 4 (𝑧 = 𝑦 → ((𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐵𝜓)))
4018, 19, 39cbvab 2891 . . 3 {𝑧 ∣ (𝑧𝑧 / 𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦𝐵𝜓)}
4111, 40eqtri 2844 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑦 ∣ (𝑦𝐵𝜓)}
42 df-rab 3135 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43 df-rab 3135 . 2 {𝑦𝐵𝜓} = {𝑦 ∣ (𝑦𝐵𝜓)}
4441, 42, 433eqtr4i 2854 1 {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  [wsb 2070  wcel 2115  {cab 2799  wnfc 2958  {crab 3130  [wsbc 3749  csb 3857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-13 2391  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-sbc 3750  df-csb 3858
This theorem is referenced by:  cbvrabv2  41547
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