Theorem ceqex 1410

Description: Equality implies equivalence with substitution.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem ceqex

Step	Hyp	Ref	Expression
1		19.8a 712	. . 3 ⊢ (x = A → ∃x x = A)
2		isset 1351	. . 3 ⊢ (A ∈ V ↔ ∃x x = A)
3	1, 2	sylibr 175	. 2 ⊢ (x = A → A ∈ V)
4		cleq2 1110	. . . 4 ⊢ (y = A → (x = y ↔ x = A))
5	4	anbi1d 469	. . . . . 6 ⊢ (y = A → ((x = y ∧ φ) ↔ (x = A ∧ φ)))
6	5	biexdv 936	. . . . 5 ⊢ (y = A → (∃x(x = y ∧ φ) ↔ ∃x(x = A ∧ φ)))
7	6	bibi2d 470	. . . 4 ⊢ (y = A → ((φ ↔ ∃x(x = y ∧ φ)) ↔ (φ ↔ ∃x(x = A ∧ φ))))
8	4, 7	imbi12d 474	. . 3 ⊢ (y = A → ((x = y → (φ ↔ ∃x(x = y ∧ φ))) ↔ (x = A → (φ ↔ ∃x(x = A ∧ φ)))))
9		19.8a 712	. . . . 5 ⊢ ((x = y ∧ φ) → ∃x(x = y ∧ φ))
10	9	exp 291	. . . 4 ⊢ (x = y → (φ → ∃x(x = y ∧ φ)))
11		ax-4 673	. . . . . 6 ⊢ (∀x(x = y → φ) → (x = y → φ))
12	11	com12 13	. . . . 5 ⊢ (x = y → (∀x(x = y → φ) → φ))
13		visset 1350	. . . . . 6 ⊢ y ∈ V
14	13	alexeq 1409	. . . . 5 ⊢ (∀x(x = y → φ) ↔ ∃x(x = y ∧ φ))
15	12, 14	syl5ibr 182	. . . 4 ⊢ (x = y → (∃x(x = y ∧ φ) → φ))
16	10, 15	impbid 397	. . 3 ⊢ (x = y → (φ ↔ ∃x(x = y ∧ φ)))
17	8, 16	vtoclg 1383	. 2 ⊢ (A ∈ V → (x = A → (φ ↔ ∃x(x = A ∧ φ))))
18	3, 17	mpcom 49	1 ⊢ (x = A → (φ ↔ ∃x(x = A ∧ φ)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  ceqsexg 1411  copsexg 1902

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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