Theorem ceqex 2857

Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)

Assertion

Ref	Expression
ceqex	⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.8a 1583	. . 3 ⊢ (𝑥 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
2		isset 2736	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylibr 133	. 2 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
4		eqeq2 2180	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
5	4	anbi1d 462	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑)))
6	5	exbidv 1818	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
7	6	bibi2d 231	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))))
8	4, 7	imbi12d 233	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))))
9		19.8a 1583	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10	9	ex 114	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
11		vex 2733	. . . . . 6 ⊢ 𝑦 ∈ V
12	11	alexeq 2856	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
13		sp 1504	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
14	13	com12 30	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
15	12, 14	syl5bir 152	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
16	10, 15	impbid 128	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
17	8, 16	vtoclg 2790	. 2 ⊢ (𝐴 ∈ V → (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))))
18	3, 17	mpcom 36	1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  ceqsexg  2858  sbc6g  2979