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Theorem bnj1468 32004
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 3802 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
2 ax-5 1904 . . 3 (𝜓 → ∀𝑦𝜓)
3 bnj1468.3 . . . . . . . 8 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
43nfcii 2970 . . . . . . 7 𝑥𝐴
54nfeq2 3000 . . . . . 6 𝑥 𝑦 = 𝐴
6 nfsbc1v 3796 . . . . . . 7 𝑥[𝑦 / 𝑥]𝜑
7 bnj1468.1 . . . . . . . 8 (𝜓 → ∀𝑥𝜓)
87nf5i 2143 . . . . . . 7 𝑥𝜓
96, 8nfbi 1897 . . . . . 6 𝑥([𝑦 / 𝑥]𝜑𝜓)
105, 9nfim 1890 . . . . 5 𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
1110nf5ri 2187 . . . 4 ((𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)) → ∀𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)))
12 ax6ev 1965 . . . . 5 𝑥 𝑥 = 𝑦
13 eqeq1 2830 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
14 bnj1468.2 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
1513, 14syl6bir 255 . . . . . 6 (𝑥 = 𝑦 → (𝑦 = 𝐴 → (𝜑𝜓)))
16 sbceq1a 3787 . . . . . . 7 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
1716bibi1d 345 . . . . . 6 (𝑥 = 𝑦 → ((𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑𝜓)))
1815, 17sylibd 240 . . . . 5 (𝑥 = 𝑦 → (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓)))
1912, 18bnj101 31879 . . . 4 𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
2011, 19bnj1131 31945 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
212, 20bnj1464 32002 . 2 (𝐴𝑉 → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑𝜓))
221, 21syl5bbr 286 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528   = wceq 1530  wcel 2107  [wsbc 3776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-sbc 3777
This theorem is referenced by:  bnj1463  32211
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