Theorem ixpeq2 8838

Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Assertion

Ref	Expression
ixpeq2	⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Proof of Theorem ixpeq2

Step	Hyp	Ref	Expression
1		ss2ixp 8837	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
2		ss2ixp 8837	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵)
3	1, 2	anim12i 613	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) → (X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶 ∧ X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵))
4		eqss 3951	. . . 4 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
5	4	ralbii 3075	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
6		r19.26 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
7	5, 6	bitri 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
8		eqss 3951	. 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶 ↔ (X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶 ∧ X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵))
9	3, 7, 8	3imtr4i 292	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∀wral 3044   ⊆ wss 3903  Xcixp 8824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-ss 3920  df-ixp 8825

This theorem is referenced by:  ixpeq2dva  8839  ixpint  8852  prdsbas3  17385  pwsbas  17391  ptbasfi  23466  ptunimpt  23480  pttopon  23481  ptcld  23498  ptrescn  23524  ptuncnv  23692  ptunhmeo  23693  ixpeq12i  36175  ptrest  37599  prdstotbnd  37774  ixpeq2d  45046  hoidmv1le  46575