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Theorem rspct 3560
Description: A closed version of rspc 3562. (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 3114 . . . 4 (∀𝑥𝐵 𝜑 ↔ ∀𝑥(𝑥𝐵𝜑))
2 eleq1 2880 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32adantr 484 . . . . . . . . 9 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → (𝑥𝐵𝐴𝐵))
4 simpr 488 . . . . . . . . 9 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → (𝜑𝜓))
53, 4imbi12d 348 . . . . . . . 8 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
65ex 416 . . . . . . 7 (𝑥 = 𝐴 → ((𝜑𝜓) → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
76a2i 14 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
87alimi 1813 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
9 nfv 1915 . . . . . . 7 𝑥 𝐴𝐵
10 rspct.1 . . . . . . 7 𝑥𝜓
119, 10nfim 1897 . . . . . 6 𝑥(𝐴𝐵𝜓)
12 nfcv 2958 . . . . . 6 𝑥𝐴
1311, 12spcgft 3538 . . . . 5 (∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))) → (𝐴𝐵 → (∀𝑥(𝑥𝐵𝜑) → (𝐴𝐵𝜓))))
148, 13syl 17 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥(𝑥𝐵𝜑) → (𝐴𝐵𝜓))))
151, 14syl7bi 258 . . 3 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑 → (𝐴𝐵𝜓))))
1615com34 91 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))))
1716pm2.43d 53 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wnf 1785  wcel 2112  wral 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446
This theorem is referenced by:  rspcdf  3561
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