Theorem bnj919 32647

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 3772	. 2 ⊢ ([𝑃 / 𝑛]𝜒 ↔ [𝑃 / 𝑛](𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj919.5	. . 3 ⊢ 𝑃 ∈ V
5		df-bnj17 32566	. . . . 5 ⊢ ((𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′) ↔ ((𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′) ∧ 𝜓′))
6		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑛 𝑃 ∈ 𝐷
7		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑛 𝐹 Fn 𝑃
8		bnj919.2	. . . . . . . 8 ⊢ (𝜑′ ↔ [𝑃 / 𝑛]𝜑)
9		nfsbc1v 3731	. . . . . . . 8 ⊢ Ⅎ𝑛[𝑃 / 𝑛]𝜑
10	8, 9	nfxfr 1856	. . . . . . 7 ⊢ Ⅎ𝑛𝜑′
11	6, 7, 10	nf3an 1905	. . . . . 6 ⊢ Ⅎ𝑛(𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′)
12		bnj919.3	. . . . . . 7 ⊢ (𝜓′ ↔ [𝑃 / 𝑛]𝜓)
13		nfsbc1v 3731	. . . . . . 7 ⊢ Ⅎ𝑛[𝑃 / 𝑛]𝜓
14	12, 13	nfxfr 1856	. . . . . 6 ⊢ Ⅎ𝑛𝜓′
15	11, 14	nfan 1903	. . . . 5 ⊢ Ⅎ𝑛((𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′) ∧ 𝜓′)
16	5, 15	nfxfr 1856	. . . 4 ⊢ Ⅎ𝑛(𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)
17		eleq1 2826	. . . . . 6 ⊢ (𝑛 = 𝑃 → (𝑛 ∈ 𝐷 ↔ 𝑃 ∈ 𝐷))
18		fneq2 6509	. . . . . . 7 ⊢ (𝑛 = 𝑃 → (𝐹 Fn 𝑛 ↔ 𝐹 Fn 𝑃))
19		sbceq1a 3722	. . . . . . . 8 ⊢ (𝑛 = 𝑃 → (𝜑 ↔ [𝑃 / 𝑛]𝜑))
20	19, 8	bitr4di 288	. . . . . . 7 ⊢ (𝑛 = 𝑃 → (𝜑 ↔ 𝜑′))
21		sbceq1a 3722	. . . . . . . 8 ⊢ (𝑛 = 𝑃 → (𝜓 ↔ [𝑃 / 𝑛]𝜓))
22	21, 12	bitr4di 288	. . . . . . 7 ⊢ (𝑛 = 𝑃 → (𝜓 ↔ 𝜓′))
23	18, 20, 22	3anbi123d 1434	. . . . . 6 ⊢ (𝑛 = 𝑃 → ((𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)))
24	17, 23	anbi12d 630	. . . . 5 ⊢ (𝑛 = 𝑃 → ((𝑛 ∈ 𝐷 ∧ (𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ (𝑃 ∈ 𝐷 ∧ (𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′))))
25		bnj252 32582	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ (𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
26		bnj252 32582	. . . . 5 ⊢ ((𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑃 ∈ 𝐷 ∧ (𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)))
27	24, 25, 26	3bitr4g 313	. . . 4 ⊢ (𝑛 = 𝑃 → ((𝑛 ∈ 𝐷 ∧ 𝐹 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′)))
28	16, 27	sbciegf 3750	. . 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 296	1 ⊢ (𝜒′ ↔ (𝑃 ∈ 𝐷 ∧ 𝐹 Fn 𝑃 ∧ 𝜑′ ∧ 𝜓′))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422  [wsbc 3711   Fn wfn 6413   ∧ w-bnj17 32565

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-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-sbc 3712  df-fn 6421  df-bnj17 32566

This theorem is referenced by:  bnj910  32828  bnj999  32838  bnj907  32847