Theorem pm13.183OLD 3662

Description: Obsolete version of pm13.183 3661 as of 29-Apr-2023. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pm13.183OLD	⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵

Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem pm13.183OLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2827	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝐵 ↔ 𝐴 = 𝐵))
2		eqeq2 2835	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐴))
3	2	bibi1d 346	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ (𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
4	3	albidv 1921	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))
5		eqeq2 2835	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
6	5	alrimiv 1928	. . 3 ⊢ (𝑦 = 𝐵 → ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
7		stdpc4 2073	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → [𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
8		sbbi 2317	. . . . 5 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵))
9		eqsb3 2941	. . . . . . 7 ⊢ ([𝑦 / 𝑧]𝑧 = 𝐵 ↔ 𝑦 = 𝐵)
10	9	bibi2i 340	. . . . . 6 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) ↔ ([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵))
11		equsb1 2530	. . . . . . 7 ⊢ [𝑦 / 𝑧]𝑧 = 𝑦
12		biimp 217	. . . . . . 7 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵) → ([𝑦 / 𝑧]𝑧 = 𝑦 → 𝑦 = 𝐵))
13	11, 12	mpi 20	. . . . . 6 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ 𝑦 = 𝐵) → 𝑦 = 𝐵)
14	10, 13	sylbi 219	. . . . 5 ⊢ (([𝑦 / 𝑧]𝑧 = 𝑦 ↔ [𝑦 / 𝑧]𝑧 = 𝐵) → 𝑦 = 𝐵)
15	8, 14	sylbi 219	. . . 4 ⊢ ([𝑦 / 𝑧](𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → 𝑦 = 𝐵)
16	7, 15	syl 17	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵) → 𝑦 = 𝐵)
17	6, 16	impbii 211	. 2 ⊢ (𝑦 = 𝐵 ↔ ∀𝑧(𝑧 = 𝑦 ↔ 𝑧 = 𝐵))
18	1, 4, 17	vtoclbg 3571	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 ↔ ∀𝑧(𝑧 = 𝐴 ↔ 𝑧 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537  [wsb 2069   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-13 2390  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-cleq 2816  df-clel 2895

This theorem is referenced by: (None)