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Theorem bnj1468 30616
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1468.1 (𝜓 → ∀𝑥𝜓)
bnj1468.2 (𝑥 = 𝐴 → (𝜑𝜓))
bnj1468.3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
bnj1468 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝜑,𝑦   𝜓,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem bnj1468
StepHypRef Expression
1 sbcco 3445 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
2 ax-5 1841 . . 3 (𝜓 → ∀𝑦𝜓)
3 bnj1468.3 . . . . . . . 8 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
43nfcii 2758 . . . . . . 7 𝑥𝐴
54nfeq2 2782 . . . . . 6 𝑥 𝑦 = 𝐴
6 nfsbc1v 3442 . . . . . . 7 𝑥[𝑦 / 𝑥]𝜑
7 bnj1468.1 . . . . . . . 8 (𝜓 → ∀𝑥𝜓)
87nf5i 2026 . . . . . . 7 𝑥𝜓
96, 8nfbi 1835 . . . . . 6 𝑥([𝑦 / 𝑥]𝜑𝜓)
105, 9nfim 1827 . . . . 5 𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
1110nf5ri 2068 . . . 4 ((𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)) → ∀𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)))
12 ax6ev 1892 . . . . 5 𝑥 𝑥 = 𝑦
13 eqeq1 2630 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
14 bnj1468.2 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
1513, 14syl6bir 244 . . . . . 6 (𝑥 = 𝑦 → (𝑦 = 𝐴 → (𝜑𝜓)))
16 sbceq1a 3433 . . . . . . 7 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
1716bibi1d 333 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))
1815, 17sylibd 229 . . . . 5 (𝑥 = 𝑦 → (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)))
1912, 18bnj101 30489 . . . 4 𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
2011, 19bnj1131 30558 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
212, 20bnj1464 30614 . 2 (𝐴𝑉 → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑𝜓))
221, 21syl5bbr 274 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478   = wceq 1480  wcel 1992  [wsbc 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-sbc 3423
This theorem is referenced by:  bnj1463  30823
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