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Theorem acexmidlemph 5867
Description: Lemma for acexmid 5873. (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 711 . . . 4 (𝜑 → (𝑥 = ∅ ∨ 𝜑))
21ralrimivw 2551 . . 3 (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
3 acexmidlem.a . . . . 5 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
43eqeq2i 2188 . . . 4 ({∅, {∅}} = 𝐴 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
5 rabid2 2653 . . . 4 ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
64, 5bitri 184 . . 3 ({∅, {∅}} = 𝐴 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
72, 6sylibr 134 . 2 (𝜑 → {∅, {∅}} = 𝐴)
8 olc 711 . . . 4 (𝜑 → (𝑥 = {∅} ∨ 𝜑))
98ralrimivw 2551 . . 3 (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
10 acexmidlem.b . . . . 5 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
1110eqeq2i 2188 . . . 4 ({∅, {∅}} = 𝐵 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
12 rabid2 2653 . . . 4 ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
1311, 12bitri 184 . . 3 ({∅, {∅}} = 𝐵 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
149, 13sylibr 134 . 2 (𝜑 → {∅, {∅}} = 𝐵)
157, 14eqtr3d 2212 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 708   = wceq 1353  wral 2455  {crab 2459  c0 3422  {csn 3592  {cpr 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-rab 2464
This theorem is referenced by:  acexmidlemab  5868
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