Theorem bnj919 34781

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

Hypotheses

Ref	Expression
bnj919.1	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj919.2	⊢ (𝜑′ ↔ [𝑃 / 𝑛]𝜑)
bnj919.3	⊢ (𝜓′ ↔ [𝑃 / 𝑛]𝜓)
bnj919.4	⊢ (𝜒′ ↔ [𝑃 / 𝑛]𝜒)
bnj919.5	⊢ 𝑃 ∈ V

Assertion

Ref	Expression
bnj919	⊢ (𝜒′ ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′))

Distinct variable groups:   𝐷,𝑛   𝑛,𝐹   𝑃,𝑛

Allowed substitution hints:   𝜑(𝑛)   𝜓(𝑛)   𝜒(𝑛)   𝜑′(𝑛)   𝜓′(𝑛)   𝜒′(𝑛)

Proof of Theorem bnj919

Step	Hyp	Ref	Expression
1		bnj919.4	. 2 ⊢ (𝜒′ ↔ [𝑃 / 𝑛]𝜒)
2		bnj919.1	. . 3 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
3	2	sbcbii 3846	. 2 ⊢ ([𝑃 / 𝑛]𝜒 ↔ [𝑃 / 𝑛](𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj919.5	. . 3 ⊢ 𝑃 ∈ V
5		df-bnj17 34701	. . . . 5 ⊢ ((𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′) ↔ ((𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′) ∧ 𝜓′))
6		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑛 𝑃 ∈ 𝐷
7		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑛 𝐹 Fn 𝑃
8		bnj919.2	. . . . . . . 8 ⊢ (𝜑′ ↔ [𝑃 / 𝑛]𝜑)
9		nfsbc1v 3808	. . . . . . . 8 ⊢ Ⅎ𝑛[𝑃 / 𝑛]𝜑
10	8, 9	nfxfr 1853	. . . . . . 7 ⊢ Ⅎ𝑛𝜑′
11	6, 7, 10	nf3an 1901	. . . . . 6 ⊢ Ⅎ𝑛(𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′)
12		bnj919.3	. . . . . . 7 ⊢ (𝜓′ ↔ [𝑃 / 𝑛]𝜓)
13		nfsbc1v 3808	. . . . . . 7 ⊢ Ⅎ𝑛[𝑃 / 𝑛]𝜓
14	12, 13	nfxfr 1853	. . . . . 6 ⊢ Ⅎ𝑛𝜓′
15	11, 14	nfan 1899	. . . . 5 ⊢ Ⅎ𝑛((𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′) ∧ 𝜓′)
16	5, 15	nfxfr 1853	. . . 4 ⊢ Ⅎ𝑛(𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)
17		eleq1 2829	. . . . . 6 ⊢ (𝑛 = 𝑃 → (𝑛 ∈ 𝐷 ↔ 𝑃 ∈ 𝐷))
18		fneq2 6660	. . . . . . 7 ⊢ (𝑛 = 𝑃 → (𝐹 Fn 𝑛 ↔ 𝐹 Fn 𝑃))
19		sbceq1a 3799	. . . . . . . 8 ⊢ (𝑛 = 𝑃 → (𝜑 ↔ [𝑃 / 𝑛]𝜑))
20	19, 8	bitr4di 289	. . . . . . 7 ⊢ (𝑛 = 𝑃 → (𝜑 ↔ 𝜑′))
21		sbceq1a 3799	. . . . . . . 8 ⊢ (𝑛 = 𝑃 → (𝜓 ↔ [𝑃 / 𝑛]𝜓))
22	21, 12	bitr4di 289	. . . . . . 7 ⊢ (𝑛 = 𝑃 → (𝜓 ↔ 𝜓′))
23	18, 20, 22	3anbi123d 1438	. . . . . 6 ⊢ (𝑛 = 𝑃 → ((𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)))
24	17, 23	anbi12d 632	. . . . 5 ⊢ (𝑛 = 𝑃 → ((𝑛 ∈ 𝐷 ∧ (𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ (𝑃 ∈ 𝐷 ∧ (𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′))))
25		bnj252 34717	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ (𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
26		bnj252 34717	. . . . 5 ⊢ ((𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑃 ∈ 𝐷 ∧ (𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)))
27	24, 25, 26	3bitr4g 314	. . . 4 ⊢ (𝑛 = 𝑃 → ((𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)))
28	16, 27	sbciegf 3827	. . 3 ⊢ (𝑃 ∈ V → ([𝑃 / 𝑛](𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)))
29	4, 28	ax-mp 5	. 2 ⊢ ([𝑃 / 𝑛](𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′))
30	1, 3, 29	3bitri 297	1 ⊢ (𝜒′ ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480  [wsbc 3788   Fn wfn 6556   ∧ w-bnj17 34700

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-sbc 3789  df-fn 6564  df-bnj17 34701

This theorem is referenced by:  bnj910  34962  bnj999  34972  bnj907  34981