Theorem acexmidlemph 5915

Description: Lemma for acexmid 5921. (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 712	. . . 4 ⊢ (𝜑 → (𝑥 = ∅ ∨ 𝜑))
2	1	ralrimivw 2571	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = ∅ ∨ 𝜑))
3		acexmidlem.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
4	3	eqeq2i 2207	. . . 4 ⊢ ({∅, {∅}} = 𝐴 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)})
5		rabid2 2674	. . . 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 712	. . . 4 ⊢ (𝜑 → (𝑥 = {∅} ∨ 𝜑))
9	8	ralrimivw 2571	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ {∅, {∅}} (𝑥 = {∅} ∨ 𝜑))
10		acexmidlem.b	. . . . 5 ⊢ 𝐵 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)}
11	10	eqeq2i 2207	. . . 4 ⊢ ({∅, {∅}} = 𝐵 ↔ {∅, {∅}} = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ 𝜑)})
12		rabid2 2674	. . . 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 2231	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∨ wo 709   = wceq 1364  ∀wral 2475  {crab 2479  ∅c0 3450  {csn 3622  {cpr 3623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-rab 2484

This theorem is referenced by:  acexmidlemab  5916