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Theorem ss2iundv 43608
Description: Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
Hypotheses
Ref Expression
ss2iundv.el ((𝜑𝑥𝐴) → 𝑌𝐶)
ss2iundv.sub ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)
ss2iundv.ss ((𝜑𝑥𝐴) → 𝐵𝐺)
Assertion
Ref Expression
ss2iundv (𝜑 𝑥𝐴 𝐵 𝑦𝐶 𝐷)
Distinct variable groups:   𝑥,𝑦,𝜑   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷   𝑦,𝐺   𝑦,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐷(𝑦)   𝐺(𝑥)   𝑌(𝑥)

Proof of Theorem ss2iundv
StepHypRef Expression
1 nfv 1910 . 2 𝑥𝜑
2 nfv 1910 . 2 𝑦𝜑
3 nfcv 2901 . 2 𝑦𝑌
4 nfcv 2901 . 2 𝑦𝐴
5 nfcv 2901 . 2 𝑦𝐵
6 nfcv 2901 . 2 𝑥𝐶
7 nfcv 2901 . 2 𝑦𝐶
8 nfcv 2901 . 2 𝑥𝐷
9 nfcv 2901 . 2 𝑦𝐺
10 ss2iundv.el . 2 ((𝜑𝑥𝐴) → 𝑌𝐶)
11 ss2iundv.sub . 2 ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)
12 ss2iundv.ss . 2 ((𝜑𝑥𝐴) → 𝐵𝐺)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12ss2iundf 43607 1 (𝜑 𝑥𝐴 𝐵 𝑦𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1535  wcel 2104  wss 3963   ciun 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-rex 3067  df-v 3479  df-ss 3980  df-iun 5000
This theorem is referenced by: (None)
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