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Theorem bj-csbsnlem 33204
Description: Lemma for bj-csbsn 33205 (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 2748 . . . 4 (𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ [𝐴 / 𝑥]𝑦 ∈ {𝑥})
2 df-sbc 3577 . . . 4 ([𝐴 / 𝑥]𝑦 ∈ {𝑥} ↔ 𝐴 ∈ {𝑥𝑦 ∈ {𝑥}})
3 clelab 2886 . . . . 5 (𝐴 ∈ {𝑥𝑦 ∈ {𝑥}} ↔ ∃𝑥(𝑥 = 𝐴𝑦 ∈ {𝑥}))
4 velsn 4337 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
54anbi2i 732 . . . . . 6 ((𝑥 = 𝐴𝑦 ∈ {𝑥}) ↔ (𝑥 = 𝐴𝑦 = 𝑥))
65exbii 1923 . . . . 5 (∃𝑥(𝑥 = 𝐴𝑦 ∈ {𝑥}) ↔ ∃𝑥(𝑥 = 𝐴𝑦 = 𝑥))
7 eqeq2 2771 . . . . . . . 8 (𝑥 = 𝐴 → (𝑦 = 𝑥𝑦 = 𝐴))
87pm5.32i 672 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝑥) ↔ (𝑥 = 𝐴𝑦 = 𝐴))
98exbii 1923 . . . . . 6 (∃𝑥(𝑥 = 𝐴𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝐴𝑦 = 𝐴))
10 19.41v 2026 . . . . . 6 (∃𝑥(𝑥 = 𝐴𝑦 = 𝐴) ↔ (∃𝑥 𝑥 = 𝐴𝑦 = 𝐴))
11 simpr 479 . . . . . . 7 ((∃𝑥 𝑥 = 𝐴𝑦 = 𝐴) → 𝑦 = 𝐴)
12 eqvisset 3351 . . . . . . . . 9 (𝑦 = 𝐴𝐴 ∈ V)
13 elisset 3355 . . . . . . . . 9 (𝐴 ∈ V → ∃𝑥 𝑥 = 𝐴)
1412, 13syl 17 . . . . . . . 8 (𝑦 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
1514ancri 576 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑥 𝑥 = 𝐴𝑦 = 𝐴))
1611, 15impbii 199 . . . . . 6 ((∃𝑥 𝑥 = 𝐴𝑦 = 𝐴) ↔ 𝑦 = 𝐴)
179, 10, 163bitri 286 . . . . 5 (∃𝑥(𝑥 = 𝐴𝑦 = 𝑥) ↔ 𝑦 = 𝐴)
183, 6, 173bitri 286 . . . 4 (𝐴 ∈ {𝑥𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
191, 2, 183bitri 286 . . 3 (𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}} ↔ 𝑦 = 𝐴)
20 df-csb 3675 . . . 4 𝐴 / 𝑥{𝑥} = {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}}
2120eleq2i 2831 . . 3 (𝑦𝐴 / 𝑥{𝑥} ↔ 𝑦 ∈ {𝑦[𝐴 / 𝑥]𝑦 ∈ {𝑥}})
22 velsn 4337 . . 3 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
2319, 21, 223bitr4i 292 . 2 (𝑦𝐴 / 𝑥{𝑥} ↔ 𝑦 ∈ {𝐴})
2423eqriv 2757 1 𝐴 / 𝑥{𝑥} = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  Vcvv 3340  [wsbc 3576  csb 3674  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-csb 3675  df-sn 4322
This theorem is referenced by:  bj-csbsn  33205
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