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Theorem bj-csbsnlem 34344
Description: Lemma for bj-csbsn 34345 (in this lemma, 𝑥 cannot occur in 𝐴). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)
Assertion
Ref Expression
bj-csbsnlem 𝐴 / 𝑥{𝑥} = {𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-csbsnlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 abid 2780 . . . 4 (𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ [𝐴 / 𝑥]𝑦 ∈ {𝑥})
2 df-sbc 3721 . . . 4 ([𝐴 / 𝑥]𝑦 ∈ {𝑥} ↔ 𝐴 ∈ {𝑥𝑦 ∈ {𝑥}})
3 clelab 2933 . . . . 5 (𝐴 ∈ {𝑥𝑦 ∈ {𝑥}} ↔ ∃𝑥(𝑥 = 𝐴𝑦 ∈ {𝑥}))
4 velsn 4541 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
54anbi2i 625 . . . . . 6 ((𝑥 = 𝐴𝑦 ∈ {𝑥}) ↔ (𝑥 = 𝐴𝑦 = 𝑥))
65exbii 1849 . . . . 5 (∃𝑥(𝑥 = 𝐴𝑦 ∈ {𝑥}) ↔ ∃𝑥(𝑥 = 𝐴𝑦 = 𝑥))
7 eqeq2 2810 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦 = 𝑥𝑦 = 𝐴))
87pm5.32i 578 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝑥) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
98exbii 1849 . . . . . 6 (∃𝑥(𝑥 = 𝐴𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝐴𝑦 = 𝐴))
10 19.41v 1950 . . . . . 6 (∃𝑥(𝑥 = 𝐴𝑦 = 𝐴) ↔ (∃𝑥 𝑥 = 𝐴𝑦 = 𝐴))
11 simpr 488 . . . . . . 7 ((∃𝑥 𝑥 = 𝐴𝑦 = 𝐴) → 𝑦 = 𝐴)
12 eqvisset 3458 . . . . . . . . 9 (𝑦 = 𝐴𝐴 ∈ V)
13 elisset 3452 . . . . . . . . 9 (𝐴 ∈ V → ∃𝑥 𝑥 = 𝐴)
1412, 13syl 17 . . . . . . . 8 (𝑦 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
1514ancri 553 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑥 𝑥 = 𝐴𝑦 = 𝐴))
1611, 15impbii 212 . . . . . 6 ((∃𝑥 𝑥 = 𝐴𝑦 = 𝐴) ↔ 𝑦 = 𝐴)
179, 10, 163bitri 300 . . . . 5 (∃𝑥(𝑥 = 𝐴𝑦 = 𝑥) ↔ 𝑦 = 𝐴)
183, 6, 173bitri 300 . . . 4 (𝐴 ∈ {𝑥𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
191, 2, 183bitri 300 . . 3 (𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
20 df-csb 3829 . . . 4 𝐴 / 𝑥{𝑥} = {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}}
2120eleq2i 2881 . . 3 (𝑦𝐴 / 𝑥{𝑥} ↔ 𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}})
22 velsn 4541 . . 3 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
2319, 21, 223bitr4i 306 . 2 (𝑦𝐴 / 𝑥{𝑥} ↔ 𝑦 ∈ {𝐴})
2423eqriv 2795 1 𝐴 / 𝑥{𝑥} = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  wcel 2111  {cab 2776  Vcvv 3441  [wsbc 3720  csb 3828  {csn 4525
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
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 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721  df-csb 3829  df-sn 4526
This theorem is referenced by:  bj-csbsn  34345
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