Theorem ixpeq2 8977

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 8976	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
2		ss2ixp 8976	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵)
3	1, 2	anim12i 625	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) → (X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶 ∧ X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵))
4		eqss 3928	. . . 4 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
5	4	ralbii 3112	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
6		r19.26 3126	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
7	5, 6	bitri 278	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
8		eqss 3928	. 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶 ↔ (X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶 ∧ X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵))
9	3, 7, 8	3imtr4i 295	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∀wral 3080   ⊆ wss 3900  Xcixp 8963

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-ss 3901  df-ixp 8964

This theorem is used by:  ixpeq2dva  8978  ixpint  8991  prdsbas3  17702  pwsbas  17708  ptbasfi  23955  ptunimpt  23969  pttopon  23970  ptcld  23987  ptrescn  24013  ptuncnv  24181  ptunhmeo  24182  ixpeq12i  37205  ptrest  38805  prdstotbnd  38994  ixpeq2d  46359  hoidmv1le  47877