Theorem pm14.122b 40753

Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Assertion

Ref	Expression
pm14.122b	⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem pm14.122b

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2833	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
2	1	imbi2d 343	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴)))
3	2	albidv 1917	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴)))
4		dfsbcq 3773	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
5	4	bibi1d 346	. . . 4 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
6	3, 5	imbi12d 347	. . 3 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑)) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑))))
7		sbc5 3799	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8		nfa1 2151	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑦)
9		simpr 487	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑)
10		ancr 549	. . . . . . 7 ⊢ ((𝜑 → 𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦 ∧ 𝜑)))
11	10	sps 2180	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → (𝑥 = 𝑦 ∧ 𝜑)))
12	9, 11	impbid2 228	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ((𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜑))
13	8, 12	exbid 2221	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥𝜑))
14	7, 13	syl5bb 285	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝜑))
15	6, 14	vtoclg 3567	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥𝜑)))
16	15	pm5.32d 579	1 ⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) ∧ ∃𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  [wsbc 3771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496  df-sbc 3772

This theorem is referenced by:  pm14.122c  40754