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Theorem acexmidlemph 5807
 Description: Lemma for acexmid 5813. (Contributed by Jim Kingdon, 6-Aug-2019.)
Hypotheses
Ref Expression
acexmidlem.a 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
acexmidlem.b 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
acexmidlem.c 𝐶 = {𝐴, 𝐵}
Assertion
Ref Expression
acexmidlemph (𝜑𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Proof of Theorem acexmidlemph
StepHypRef Expression
1 olc 701 . . . 4 (𝜑 → (𝑥 = ∅ ∨ 𝜑))
21ralrimivw 2528 . . 3 (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
3 acexmidlem.a . . . . 5 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
43eqeq2i 2165 . . . 4 ({∅, {∅}} = 𝐴 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
5 rabid2 2630 . . . 4 ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
64, 5bitri 183 . . 3 ({∅, {∅}} = 𝐴 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
72, 6sylibr 133 . 2 (𝜑 → {∅, {∅}} = 𝐴)
8 olc 701 . . . 4 (𝜑 → (𝑥 = {∅} ∨ 𝜑))
98ralrimivw 2528 . . 3 (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
10 acexmidlem.b . . . . 5 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
1110eqeq2i 2165 . . . 4 ({∅, {∅}} = 𝐵 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
12 rabid2 2630 . . . 4 ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
1311, 12bitri 183 . . 3 ({∅, {∅}} = 𝐵 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
149, 13sylibr 133 . 2 (𝜑 → {∅, {∅}} = 𝐵)
157, 14eqtr3d 2189 1 (𝜑𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 698   = wceq 1332  ∀wral 2432  {crab 2436  ∅c0 3390  {csn 3556  {cpr 3557 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-ral 2437  df-rab 2441 This theorem is referenced by:  acexmidlemab  5808
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