Theorem pm13.192 44986

Description: Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)

Assertion

Ref	Expression
pm13.192	⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem pm13.192

Step	Hyp	Ref	Expression
1		biimpr 222	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝑥 = 𝐴))
2	1	alimi 1831	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝐴))
3		eqeq1 2766	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
4	3	equsalvw 2024	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝐴) ↔ 𝑦 = 𝐴)
5	2, 4	sylib 220	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) → 𝑦 = 𝐴)
6		eqeq2 2774	. . . . . . 7 ⊢ (𝐴 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑥 = 𝑦))
7	6	eqcoms 2770	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝐴 ↔ 𝑥 = 𝑦))
8	7	alrimiv 1947	. . . . 5 ⊢ (𝑦 = 𝐴 → ∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦))
9	5, 8	impbii 211	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ↔ 𝑦 = 𝐴)
10	9	anbi1i 633	. . 3 ⊢ ((∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ (𝑦 = 𝐴 ∧ 𝜑))
11	10	exbii 1868	. 2 ⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
12		sbc5 3772	. 2 ⊢ ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
13	11, 12	bitr4i 280	1 ⊢ (∃𝑦(∀𝑥(𝑥 = 𝐴 ↔ 𝑥 = 𝑦) ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560  ∃wex 1799  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3745

This theorem is referenced by: (None)