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Theorem acexmidlemph 5733
Description: Lemma for acexmid 5739. (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 683 . . . 4 (𝜑 → (𝑥 = ∅ ∨ 𝜑))
21ralrimivw 2481 . . 3 (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
3 acexmidlem.a . . . . 5 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
43eqeq2i 2126 . . . 4 ({∅, {∅}} = 𝐴 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
5 rabid2 2582 . . . 4 ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
64, 5bitri 183 . . 3 ({∅, {∅}} = 𝐴 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
72, 6sylibr 133 . 2 (𝜑 → {∅, {∅}} = 𝐴)
8 olc 683 . . . 4 (𝜑 → (𝑥 = {∅} ∨ 𝜑))
98ralrimivw 2481 . . 3 (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
10 acexmidlem.b . . . . 5 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
1110eqeq2i 2126 . . . 4 ({∅, {∅}} = 𝐵 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
12 rabid2 2582 . . . 4 ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
1311, 12bitri 183 . . 3 ({∅, {∅}} = 𝐵 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
149, 13sylibr 133 . 2 (𝜑 → {∅, {∅}} = 𝐵)
157, 14eqtr3d 2150 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 680   = wceq 1314  wral 2391  {crab 2395  c0 3331  {csn 3495  {cpr 3496
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2396  df-rab 2400
This theorem is referenced by:  acexmidlemab  5734
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