Theorem opeq2OLD 4757

Description: Obsolete version of opeq2 4756 as of 25-May-2024. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
opeq2OLD	⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Proof of Theorem opeq2OLD

Step	Hyp	Ref	Expression
1		eleq1 2838	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
2	1	anbi2d 632	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐶 ∈ V ∧ 𝐴 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐵 ∈ V)))
3		preq2 4620	. . . 4 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
4	3	preq2d 4626	. . 3 ⊢ (𝐴 = 𝐵 → {{𝐶}, {𝐶, 𝐴}} = {{𝐶}, {𝐶, 𝐵}})
5	2, 4	ifbieq1d 4437	. 2 ⊢ (𝐴 = 𝐵 → if((𝐶 ∈ V ∧ 𝐴 ∈ V), {{𝐶}, {𝐶, 𝐴}}, ∅) = if((𝐶 ∈ V ∧ 𝐵 ∈ V), {{𝐶}, {𝐶, 𝐵}}, ∅))
6		dfopif 4750	. 2 ⊢ ⟨𝐶, 𝐴⟩ = if((𝐶 ∈ V ∧ 𝐴 ∈ V), {{𝐶}, {𝐶, 𝐴}}, ∅)
7		dfopif 4750	. 2 ⊢ ⟨𝐶, 𝐵⟩ = if((𝐶 ∈ V ∧ 𝐵 ∈ V), {{𝐶}, {𝐶, 𝐵}}, ∅)
8	5, 6, 7	3eqtr4g 2819	1 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  Vcvv 3407  ∅c0 4221  ifcif 4413  {csn 4515  {cpr 4517  ⟨cop 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3077  df-v 3409  df-dif 3857  df-un 3859  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522

This theorem is referenced by: (None)