Theorem iineq2d 3918

Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)

Hypotheses

Ref	Expression
iineq2d.1	⊢ Ⅎ𝑥𝜑
iineq2d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
iineq2d	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Proof of Theorem iineq2d

Step	Hyp	Ref	Expression
1		iineq2d.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		iineq2d.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
3	2	ex 115	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶))
4	1, 3	ralrimi 2558	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
5		iineq2 3915	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)
6	4, 5	syl 14	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363  Ⅎwnf 1470   ∈ wcel 2158  ∀wral 2465  ∩ ciin 3899

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-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-ral 2470  df-iin 3901

This theorem is referenced by:  iineq2dv  3920