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Theorem rspct 2737
Description: A closed version of rspc 2738. (Contributed by Andrew Salmon, 6-Jun-2011.)
Hypothesis
Ref Expression
rspct.1 𝑥𝜓
Assertion
Ref Expression
rspct (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rspct
StepHypRef Expression
1 df-ral 2380 . . . 4 (∀𝑥𝐵 𝜑 ↔ ∀𝑥(𝑥𝐵𝜑))
2 eleq1 2162 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32adantr 272 . . . . . . . . 9 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → (𝑥𝐵𝐴𝐵))
4 simpr 109 . . . . . . . . 9 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → (𝜑𝜓))
53, 4imbi12d 233 . . . . . . . 8 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
65ex 114 . . . . . . 7 (𝑥 = 𝐴 → ((𝜑𝜓) → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
76a2i 11 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
87alimi 1399 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
9 nfv 1476 . . . . . . 7 𝑥 𝐴𝐵
10 rspct.1 . . . . . . 7 𝑥𝜓
119, 10nfim 1519 . . . . . 6 𝑥(𝐴𝐵𝜓)
12 nfcv 2240 . . . . . 6 𝑥𝐴
1311, 12spcgft 2718 . . . . 5 (∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))) → (𝐴𝐵 → (∀𝑥(𝑥𝐵𝜑) → (𝐴𝐵𝜓))))
148, 13syl 14 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥(𝑥𝐵𝜑) → (𝐴𝐵𝜓))))
151, 14syl7bi 164 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))))
1615com34 83 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))))
1716pm2.43d 50 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1297   = wceq 1299  wnf 1404  wcel 1448  wral 2375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643
This theorem is referenced by:  sumdc2  12587
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