Theorem bnj89 32600

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj89.1	⊢ 𝑍 ∈ V

Assertion

Ref	Expression
bnj89	⊢ ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝑍   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑍(𝑦)

Proof of Theorem bnj89

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex2 3777	. . 3 ⊢ ([𝑍 / 𝑦]∃𝑤∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∃𝑤[𝑍 / 𝑦]∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
2		sbcal 3776	. . . 4 ⊢ ([𝑍 / 𝑦]∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑥[𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤))
3	2	exbii 1851	. . 3 ⊢ (∃𝑤[𝑍 / 𝑦]∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∃𝑤∀𝑥[𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤))
4		bnj89.1	. . . . . . 7 ⊢ 𝑍 ∈ V
5		sbcbig 3765	. . . . . . 7 ⊢ (𝑍 ∈ V → ([𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑 ↔ [𝑍 / 𝑦]𝑥 = 𝑤)))
6	4, 5	ax-mp 5	. . . . . 6 ⊢ ([𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑 ↔ [𝑍 / 𝑦]𝑥 = 𝑤))
7		sbcg 3791	. . . . . . . 8 ⊢ (𝑍 ∈ V → ([𝑍 / 𝑦]𝑥 = 𝑤 ↔ 𝑥 = 𝑤))
8	4, 7	ax-mp 5	. . . . . . 7 ⊢ ([𝑍 / 𝑦]𝑥 = 𝑤 ↔ 𝑥 = 𝑤)
9	8	bibi2i 337	. . . . . 6 ⊢ (([𝑍 / 𝑦]𝜑 ↔ [𝑍 / 𝑦]𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
10	6, 9	bitri 274	. . . . 5 ⊢ ([𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
11	10	albii 1823	. . . 4 ⊢ (∀𝑥[𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑥([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
12	11	exbii 1851	. . 3 ⊢ (∃𝑤∀𝑥[𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤) ↔ ∃𝑤∀𝑥([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
13	1, 3, 12	3bitri 296	. 2 ⊢ ([𝑍 / 𝑦]∃𝑤∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∃𝑤∀𝑥([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
14		eu6 2574	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑤∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
15	14	sbcbii 3772	. 2 ⊢ ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ [𝑍 / 𝑦]∃𝑤∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
16		eu6 2574	. 2 ⊢ (∃!𝑥[𝑍 / 𝑦]𝜑 ↔ ∃𝑤∀𝑥([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
17	13, 15, 16	3bitr4i 302	1 ⊢ ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1537  ∃wex 1783   ∈ wcel 2108  ∃!weu 2568  Vcvv 3422  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-sbc 3712

This theorem is referenced by:  bnj130  32754  bnj207  32761