Theorem preqsndOLD 4610

Description: Obsolete version of preqsnd 4609 as of 12-Jun-2022: Hypothesis preqsndOLD.3 is not needed. (Contributed by Thierry Arnoux, 27-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
preqsnd.1	⊢ (𝜑 → 𝐴 ∈ V)
preqsnd.2	⊢ (𝜑 → 𝐵 ∈ V)
preqsndOLD.3	⊢ (𝜑 → 𝐶 ∈ V)

Assertion

Ref	Expression
preqsndOLD	⊢ (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))

Proof of Theorem preqsndOLD

Step	Hyp	Ref	Expression
1		preqsnd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
2		preqsnd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
3		preqsndOLD.3	. 2 ⊢ (𝜑 → 𝐶 ∈ V)
4		dfsn2 4412	. . . 4 ⊢ {𝐶} = {𝐶, 𝐶}
5	4	eqeq2i 2837	. . 3 ⊢ ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
6		preq12bg 4603	. . . 4 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))))
7		oridm 933	. . . 4 ⊢ (((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶))
8	6, 7	syl6bb 279	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
9	5, 8	syl5bb 275	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐶 ∈ V ∧ 𝐶 ∈ V)) → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
10	1, 2, 3, 3, 9	syl22anc 872	1 ⊢ (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 878   = wceq 1656   ∈ wcel 2164  Vcvv 3414  {csn 4399  {cpr 4401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-un 3803  df-sn 4400  df-pr 4402

This theorem is referenced by: (None)