Theorem ceqex 2671

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 1482	. . 3 ⊢ (𝑥 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
2		isset 2561	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylibr 137	. 2 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
4		eqeq2 2049	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
5	4	anbi1d 438	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑)))
6	5	exbidv 1706	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
7	6	bibi2d 221	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))))
8	4, 7	imbi12d 223	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))))
9		19.8a 1482	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10	9	ex 108	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
11		vex 2560	. . . . . 6 ⊢ 𝑦 ∈ V
12	11	alexeq 2670	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
13		sp 1401	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
14	13	com12 27	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
15	12, 14	syl5bir 142	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
16	10, 15	impbid 120	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
17	8, 16	vtoclg 2613	. 2 ⊢ (𝐴 ∈ V → (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))))
18	3, 17	mpcom 32	1 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  ceqsexg  2672  sbc6g  2788