Theorem br1steqgOLD 31674

Description: Obsolete version of br1steqg 31672 as of 9-Feb-2022. (Contributed by Scott Fenton, 2-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
br1steqgOLD	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴))

Proof of Theorem br1steqgOLD

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4402	. . . . . 6 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	breq1d 4663	. . . . 5 ⊢ (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩1st 𝐶 ↔ ⟨𝐴, 𝑦⟩1st 𝐶))
3		eqeq2 2633	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐶 = 𝑥 ↔ 𝐶 = 𝐴))
4	2, 3	bibi12d 335	. . . 4 ⊢ (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩1st 𝐶 ↔ 𝐶 = 𝑥) ↔ (⟨𝐴, 𝑦⟩1st 𝐶 ↔ 𝐶 = 𝐴)))
5	4	imbi2d 330	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐶 ∈ 𝑋 → (⟨𝑥, 𝑦⟩1st 𝐶 ↔ 𝐶 = 𝑥)) ↔ (𝐶 ∈ 𝑋 → (⟨𝐴, 𝑦⟩1st 𝐶 ↔ 𝐶 = 𝐴))))
6		opeq2 4403	. . . . . 6 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
7	6	breq1d 4663	. . . . 5 ⊢ (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩1st 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶))
8	7	bibi1d 333	. . . 4 ⊢ (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩1st 𝐶 ↔ 𝐶 = 𝐴) ↔ (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴)))
9	8	imbi2d 330	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐶 ∈ 𝑋 → (⟨𝐴, 𝑦⟩1st 𝐶 ↔ 𝐶 = 𝐴)) ↔ (𝐶 ∈ 𝑋 → (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴))))
10		breq2 4657	. . . 4 ⊢ (𝑧 = 𝐶 → (⟨𝑥, 𝑦⟩1st 𝑧 ↔ ⟨𝑥, 𝑦⟩1st 𝐶))
11		eqeq1 2626	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 = 𝑥 ↔ 𝐶 = 𝑥))
12		vex 3203	. . . . 5 ⊢ 𝑥 ∈ V
13		vex 3203	. . . . 5 ⊢ 𝑦 ∈ V
14	12, 13	br1steq 31670	. . . 4 ⊢ (⟨𝑥, 𝑦⟩1st 𝑧 ↔ 𝑧 = 𝑥)
15	10, 11, 14	vtoclbg 3267	. . 3 ⊢ (𝐶 ∈ 𝑋 → (⟨𝑥, 𝑦⟩1st 𝐶 ↔ 𝐶 = 𝑥))
16	5, 9, 15	vtocl2g 3270	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 𝑋 → (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴)))
17	16	3impia 1261	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ⟨cop 4183   class class class wbr 4653  1^st c1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168

This theorem is referenced by: (None)