Theorem bnj89 34718

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 3817	. . 3 ⊢ ([𝑍 / 𝑦]∃𝑤∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∃𝑤[𝑍 / 𝑦]∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
2		sbcal 3816	. . . 4 ⊢ ([𝑍 / 𝑦]∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑥[𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤))
3	2	exbii 1848	. . 3 ⊢ (∃𝑤[𝑍 / 𝑦]∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∃𝑤∀𝑥[𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤))
4		bnj89.1	. . . . . . 7 ⊢ 𝑍 ∈ V
5		sbcbig 3808	. . . . . . 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 3829	. . . . . . . 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 275	. . . . 5 ⊢ ([𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤) ↔ ([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
11	10	albii 1819	. . . 4 ⊢ (∀𝑥[𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑥([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
12	11	exbii 1848	. . 3 ⊢ (∃𝑤∀𝑥[𝑍 / 𝑦](𝜑 ↔ 𝑥 = 𝑤) ↔ ∃𝑤∀𝑥([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
13	1, 3, 12	3bitri 297	. 2 ⊢ ([𝑍 / 𝑦]∃𝑤∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∃𝑤∀𝑥([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
14		eu6 2568	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑤∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
15	14	sbcbii 3813	. 2 ⊢ ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ [𝑍 / 𝑦]∃𝑤∀𝑥(𝜑 ↔ 𝑥 = 𝑤))
16		eu6 2568	. 2 ⊢ (∃!𝑥[𝑍 / 𝑦]𝜑 ↔ ∃𝑤∀𝑥([𝑍 / 𝑦]𝜑 ↔ 𝑥 = 𝑤))
17	13, 15, 16	3bitr4i 303	1 ⊢ ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1538  ∃wex 1779   ∈ wcel 2109  ∃!weu 2562  Vcvv 3450  [wsbc 3756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-sbc 3757

This theorem is referenced by:  bnj130  34871  bnj207  34878