Theorem iotavalb 38631

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5862. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotavalb	⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotavalb

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 5862	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2		iotasbc 38620	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦)))
3		iotaexeu 38619	. . . . 5 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
4		eqsbc3 3475	. . . . 5 ⊢ ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
5	3, 4	syl 17	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦))
6	2, 5	bitr3d 270	. . 3 ⊢ (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))
7		equequ2 1953	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	bibi2d 332	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
9	8	albidv 1849	. . . . 5 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	9	biimpac 503	. . . 4 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11	10	exlimiv 1858	. . 3 ⊢ (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
12	6, 11	syl6bir 244	. 2 ⊢ (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13	1, 12	impbid2 216	1 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  Vcvv 3200  [wsbc 3435  ℩cio 5849

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by:  iotavalsb  38634