Theorem iineq1 3880

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
iineq1	⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iineq1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 2661	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
2	1	abbidv 2284	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶})
3		df-iin 3869	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
4		df-iin 3869	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}
5	2, 3, 4	3eqtr4g 2224	1 ⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∩ ciin 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-iin 3869

This theorem is referenced by:  riin0  3937  iin0r  4148