Theorem iineq1 3941

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 2702	. . 3 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
2	1	abbidv 2323	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶})
3		df-iin 3930	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶}
4		df-iin 3930	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}
5	2, 3, 4	3eqtr4g 2263	1 ⊢ (𝐴 = 𝐵 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2176  {cab 2191  ∀wral 2484  ∩ ciin 3928

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-iin 3930

This theorem is referenced by:  riin0  3999  iin0r  4213