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Theorem ssiinf 3830
Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
ssiinf.1 𝑥𝐶
Assertion
Ref Expression
ssiinf (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)

Proof of Theorem ssiinf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 2661 . . . . 5 𝑦 ∈ V
2 eliin 3786 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
31, 2ax-mp 5 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
43ralbii 2416 . . 3 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶𝑥𝐴 𝑦𝐵)
5 ssiinf.1 . . . 4 𝑥𝐶
6 nfcv 2256 . . . 4 𝑦𝐴
75, 6ralcomf 2567 . . 3 (∀𝑦𝐶𝑥𝐴 𝑦𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
84, 7bitri 183 . 2 (∀𝑦𝐶 𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
9 dfss3 3055 . 2 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑦𝐶 𝑦 𝑥𝐴 𝐵)
10 dfss3 3055 . . 3 (𝐶𝐵 ↔ ∀𝑦𝐶 𝑦𝐵)
1110ralbii 2416 . 2 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑥𝐴𝑦𝐶 𝑦𝐵)
128, 9, 113bitr4i 211 1 (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1463  wnfc 2243  wral 2391  Vcvv 2658  wss 3039   ciin 3782
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-in 3045  df-ss 3052  df-iin 3784
This theorem is referenced by:  ssiin  3831  dmiin  4753
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