Theorem acexmidlemph 5868

Description: Lemma for acexmid 5874. (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	⊢ (𝜑 → 𝐴 = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Proof of Theorem acexmidlemph

Step	Hyp	Ref	Expression
1		olc 711	. . . 4 ⊢ (𝜑 → (𝑥 = ∅ ∨ 𝜑))
2	1	ralrimivw 2551	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
3		acexmidlem.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
4	3	eqeq2i 2188	. . . 4 ⊢ ({∅, {∅}} = 𝐴 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
5		rabid2 2654	. . . 4 ⊢ ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
6	4, 5	bitri 184	. . 3 ⊢ ({∅, {∅}} = 𝐴 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
7	2, 6	sylibr 134	. 2 ⊢ (𝜑 → {∅, {∅}} = 𝐴)
8		olc 711	. . . 4 ⊢ (𝜑 → (𝑥 = {∅} ∨ 𝜑))
9	8	ralrimivw 2551	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
10		acexmidlem.b	. . . . 5 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
11	10	eqeq2i 2188	. . . 4 ⊢ ({∅, {∅}} = 𝐵 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
12		rabid2 2654	. . . 4 ⊢ ({∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)} ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
13	11, 12	bitri 184	. . 3 ⊢ ({∅, {∅}} = 𝐵 ↔ ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
14	9, 13	sylibr 134	. 2 ⊢ (𝜑 → {∅, {∅}} = 𝐵)
15	7, 14	eqtr3d 2212	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∨ wo 708   = wceq 1353  ∀wral 2455  {crab 2459  ∅c0 3423  {csn 3593  {cpr 3594

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  5869