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Theorem bj-csbsnlem 33206
Description: Lemma for bj-csbsn 33207 (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 2794 . . . 4 (𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ [𝐴 / 𝑥]𝑦 ∈ {𝑥})
2 df-sbc 3634 . . . 4 ([𝐴 / 𝑥]𝑦 ∈ {𝑥} ↔ 𝐴 ∈ {𝑥𝑦 ∈ {𝑥}})
3 clelab 2932 . . . . 5 (𝐴 ∈ {𝑥𝑦 ∈ {𝑥}} ↔ ∃𝑥(𝑥 = 𝐴𝑦 ∈ {𝑥}))
4 velsn 4386 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
54anbi2i 611 . . . . . 6 ((𝑥 = 𝐴𝑦 ∈ {𝑥}) ↔ (𝑥 = 𝐴𝑦 = 𝑥))
65exbii 1933 . . . . 5 (∃𝑥(𝑥 = 𝐴𝑦 ∈ {𝑥}) ↔ ∃𝑥(𝑥 = 𝐴𝑦 = 𝑥))
7 eqeq2 2817 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦 = 𝑥𝑦 = 𝐴))
87pm5.32i 566 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝑥) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
98exbii 1933 . . . . . 6 (∃𝑥(𝑥 = 𝐴𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝐴𝑦 = 𝐴))
10 19.41v 2040 . . . . . 6 (∃𝑥(𝑥 = 𝐴𝑦 = 𝐴) ↔ (∃𝑥 𝑥 = 𝐴𝑦 = 𝐴))
11 simpr 473 . . . . . . 7 ((∃𝑥 𝑥 = 𝐴𝑦 = 𝐴) → 𝑦 = 𝐴)
12 eqvisset 3405 . . . . . . . . 9 (𝑦 = 𝐴𝐴 ∈ V)
13 elisset 3409 . . . . . . . . 9 (𝐴 ∈ V → ∃𝑥 𝑥 = 𝐴)
1412, 13syl 17 . . . . . . . 8 (𝑦 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
1514ancri 541 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑥 𝑥 = 𝐴𝑦 = 𝐴))
1611, 15impbii 200 . . . . . 6 ((∃𝑥 𝑥 = 𝐴𝑦 = 𝐴) ↔ 𝑦 = 𝐴)
179, 10, 163bitri 288 . . . . 5 (∃𝑥(𝑥 = 𝐴𝑦 = 𝑥) ↔ 𝑦 = 𝐴)
183, 6, 173bitri 288 . . . 4 (𝐴 ∈ {𝑥𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
191, 2, 183bitri 288 . . 3 (𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
20 df-csb 3729 . . . 4 𝐴 / 𝑥{𝑥} = {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}}
2120eleq2i 2877 . . 3 (𝑦𝐴 / 𝑥{𝑥} ↔ 𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}})
22 velsn 4386 . . 3 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
2319, 21, 223bitr4i 294 . 2 (𝑦𝐴 / 𝑥{𝑥} ↔ 𝑦 ∈ {𝐴})
2423eqriv 2803 1 𝐴 / 𝑥{𝑥} = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wex 1859  wcel 2156  {cab 2792  Vcvv 3391  [wsbc 3633  csb 3728  {csn 4370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-v 3393  df-sbc 3634  df-csb 3729  df-sn 4371
This theorem is referenced by:  bj-csbsn  33207
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