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Theorem csbcnvgALT 5898
Description: Move class substitution in and out of the converse of a relation. Version of csbcnv 5897 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbcnvgALT (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)

Proof of Theorem csbcnvgALT
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbr123 5202 . . . . 5 ([𝐴 / 𝑥]𝑧𝐹𝑦𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦)
2 csbconstg 3927 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
3 csbconstg 3927 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
42, 3breq12d 5161 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝑧𝐴 / 𝑥𝐹𝑦))
51, 4bitrid 283 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦𝑧𝐴 / 𝑥𝐹𝑦))
65opabbidv 5214 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
7 csbopabgALT 5566 . . 3 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8 df-cnv 5697 . . . 4 𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
98a1i 11 . . 3 (𝐴𝑉𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
106, 7, 93eqtr4rd 2786 . 2 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11 df-cnv 5697 . . 3 𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1211csbeq2i 3916 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1310, 12eqtr4di 2793 1 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  [wsbc 3791  csb 3908   class class class wbr 5148  {copab 5210  ccnv 5688
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-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697
This theorem is referenced by: (None)
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