Theorem csbcnvgALTOLD 5873

Description: Obsolete version of csbcnv 5871 as of 4-Jun-2026. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbcnvgALTOLD	⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)

Proof of Theorem csbcnvgALTOLD

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcbr123 5164	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦)
2		csbconstg 3871	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
3		csbconstg 3871	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
4	2, 3	breq12d 5121	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦))
5	1, 4	bitrid 286	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦))
6	5	opabbidv 5176	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦})
7		csbopabw 5540	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8		df-cnv 5668	. . . 4 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}
9	8	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦})
10	6, 7, 9	3eqtr4rd 2808	. 2 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11		df-cnv 5668	. . 3 ⊢ ◡𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
12	11	csbeq2i 3860	. 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
13	10, 12	eqtr4di 2815	1 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  [wsbc 3743  ⦋csb 3852   class class class wbr 5108  {copab 5172  ^◡ccnv 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-cnv 5668

This theorem is used by: (None)