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Theorem rspct 3520
Description: A closed version of rspc 3522. (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 3086 . . . 4 (∀𝑥𝐵 𝜑 ↔ ∀𝑥(𝑥𝐵𝜑))
2 eleq1 2846 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32adantr 473 . . . . . . . . 9 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → (𝑥𝐵𝐴𝐵))
4 simpr 477 . . . . . . . . 9 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → (𝜑𝜓))
53, 4imbi12d 337 . . . . . . . 8 ((𝑥 = 𝐴 ∧ (𝜑𝜓)) → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
65ex 405 . . . . . . 7 (𝑥 = 𝐴 → ((𝜑𝜓) → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
76a2i 14 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
87alimi 1775 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))))
9 nfv 1874 . . . . . . 7 𝑥 𝐴𝐵
10 rspct.1 . . . . . . 7 𝑥𝜓
119, 10nfim 1860 . . . . . 6 𝑥(𝐴𝐵𝜓)
12 nfcv 2925 . . . . . 6 𝑥𝐴
1311, 12spcgft 3499 . . . . 5 (∀𝑥(𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓))) → (𝐴𝐵 → (∀𝑥(𝑥𝐵𝜑) → (𝐴𝐵𝜓))))
148, 13syl 17 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥(𝑥𝐵𝜑) → (𝐴𝐵𝜓))))
151, 14syl7bi 247 . . 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 198  wa 387  wal 1506   = wceq 1508  wnf 1747  wcel 2051  wral 3081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-v 3410
This theorem is referenced by:  rspcdf  3521
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