Theorem intmin4 2554

Description: Elimination of a conjunct in a class intersection.

Assertion

Ref	Expression
intmin4	⊢ (A ⊆ ∩{x∣φ} → ∩{x∣(A ⊆ x ⋀ φ)} = ∩{x∣φ})

Distinct variable group:   x,A

Proof of Theorem intmin4

Step	Hyp	Ref	Expression
1		ssintab 2545	. . . 4 ⊢ (A ⊆ ∩{x∣φ} ↔ ∀x(φ → A ⊆ x))
2		pm3.27 323	. . . . . . . 8 ⊢ ((A ⊆ x ⋀ φ) → φ)
3		ancr 295	. . . . . . . 8 ⊢ ((φ → A ⊆ x) → (φ → (A ⊆ x ⋀ φ)))
4	2, 3	impbid2 517	. . . . . . 7 ⊢ ((φ → A ⊆ x) → ((A ⊆ x ⋀ φ) ↔ φ))
5	4	imbi1d 612	. . . . . 6 ⊢ ((φ → A ⊆ x) → (((A ⊆ x ⋀ φ) → y ∈ x) ↔ (φ → y ∈ x)))
6	5	19.20i 990	. . . . 5 ⊢ (∀x(φ → A ⊆ x) → ∀x(((A ⊆ x ⋀ φ) → y ∈ x) ↔ (φ → y ∈ x)))
7		19.15 995	. . . . 5 ⊢ (∀x(((A ⊆ x ⋀ φ) → y ∈ x) ↔ (φ → y ∈ x)) → (∀x((A ⊆ x ⋀ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
8	6, 7	syl 10	. . . 4 ⊢ (∀x(φ → A ⊆ x) → (∀x((A ⊆ x ⋀ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
9	1, 8	sylbi 199	. . 3 ⊢ (A ⊆ ∩{x∣φ} → (∀x((A ⊆ x ⋀ φ) → y ∈ x) ↔ ∀x(φ → y ∈ x)))
10		visset 1809	. . . 4 ⊢ y ∈ V
11	10	elintab 2539	. . 3 ⊢ (y ∈ ∩{x∣(A ⊆ x ⋀ φ)} ↔ ∀x((A ⊆ x ⋀ φ) → y ∈ x))
12	10	elintab 2539	. . 3 ⊢ (y ∈ ∩{x∣φ} ↔ ∀x(φ → y ∈ x))
13	9, 11, 12	3bitr4g 554	. 2 ⊢ (A ⊆ ∩{x∣φ} → (y ∈ ∩{x∣(A ⊆ x ⋀ φ)} ↔ y ∈ ∩{x∣φ}))
14	13	eqrdv 1471	1 ⊢ (A ⊆ ∩{x∣φ} → ∩{x∣(A ⊆ x ⋀ φ)} = ∩{x∣φ})

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  {cab 1461   ⊆ wss 2043  ∩cint 2528

This theorem is referenced by:  abfii3 4543

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-in 2047  df-ss 2049  df-int 2529

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