Theorem ceqex 2865

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 1590	. . 3 ⊢ (𝑥 = 𝐴 → ∃𝑥 𝑥 = 𝐴)
2		isset 2744	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3	1, 2	sylibr 134	. 2 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
4		eqeq2 2187	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
5	4	anbi1d 465	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑)))
6	5	exbidv 1825	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
7	6	bibi2d 232	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))))
8	4, 7	imbi12d 234	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))))
9		19.8a 1590	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10	9	ex 115	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
11		vex 2741	. . . . . 6 ⊢ 𝑦 ∈ V
12	11	alexeq 2864	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
13		sp 1511	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
14	13	com12 30	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
15	12, 14	biimtrrid 153	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
16	10, 15	impbid 129	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
17	8, 16	vtoclg 2798	. 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 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740

This theorem is referenced by:  ceqsexg  2866  sbc6g  2988