Theorem bnj976 32049

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

Hypotheses

Ref	Expression
bnj976.1	⊢ (𝜒 ↔ (𝑁 ∈ 𝐷 ∧ 𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓))
bnj976.2	⊢ (𝜑′ ↔ [𝐺 / 𝑓]𝜑)
bnj976.3	⊢ (𝜓′ ↔ [𝐺 / 𝑓]𝜓)
bnj976.4	⊢ (𝜒′ ↔ [𝐺 / 𝑓]𝜒)
bnj976.5	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
bnj976	⊢ (𝜒′ ↔ (𝑁 ∈ 𝐷 ∧ 𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′))

Distinct variable groups:   𝐷,𝑓   𝑓,𝑁

Allowed substitution hints:   𝜑(𝑓)   𝜓(𝑓)   𝜒(𝑓)   𝐺(𝑓)   𝜑′(𝑓)   𝜓′(𝑓)   𝜒′(𝑓)

Proof of Theorem bnj976

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj976.4	. 2 ⊢ (𝜒′ ↔ [𝐺 / 𝑓]𝜒)
2		sbccow 3795	. 2 ⊢ ([𝐺 / ℎ][ℎ / 𝑓]𝜒 ↔ [𝐺 / 𝑓]𝜒)
3		bnj976.5	. . 3 ⊢ 𝐺 ∈ V
4		bnj252 31973	. . . . . 6 ⊢ ((𝑁 ∈ 𝐷 ∧ 𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓) ↔ (𝑁 ∈ 𝐷 ∧ (𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓)))
5	4	sbcbii 3829	. . . . 5 ⊢ ([ℎ / 𝑓](𝑁 ∈ 𝐷 ∧ 𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓) ↔ [ℎ / 𝑓](𝑁 ∈ 𝐷 ∧ (𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓)))
6		bnj976.1	. . . . . 6 ⊢ (𝜒 ↔ (𝑁 ∈ 𝐷 ∧ 𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓))
7	6	sbcbii 3829	. . . . 5 ⊢ ([ℎ / 𝑓]𝜒 ↔ [ℎ / 𝑓](𝑁 ∈ 𝐷 ∧ 𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓))
8		vex 3497	. . . . . . . 8 ⊢ ℎ ∈ V
9	8	bnj525 32009	. . . . . . 7 ⊢ ([ℎ / 𝑓]𝑁 ∈ 𝐷 ↔ 𝑁 ∈ 𝐷)
10		sbc3an 3838	. . . . . . . 8 ⊢ ([ℎ / 𝑓](𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓) ↔ ([ℎ / 𝑓]𝑓 Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓))
11		bnj62 31990	. . . . . . . . 9 ⊢ ([ℎ / 𝑓]𝑓 Fn 𝑁 ↔ ℎ Fn 𝑁)
12	11	3anbi1i 1153	. . . . . . . 8 ⊢ (([ℎ / 𝑓]𝑓 Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓) ↔ (ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓))
13	10, 12	bitri 277	. . . . . . 7 ⊢ ([ℎ / 𝑓](𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓) ↔ (ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓))
14	9, 13	anbi12i 628	. . . . . 6 ⊢ (([ℎ / 𝑓]𝑁 ∈ 𝐷 ∧ [ℎ / 𝑓](𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓)) ↔ (𝑁 ∈ 𝐷 ∧ (ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓)))
15		sbcan 3821	. . . . . 6 ⊢ ([ℎ / 𝑓](𝑁 ∈ 𝐷 ∧ (𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓)) ↔ ([ℎ / 𝑓]𝑁 ∈ 𝐷 ∧ [ℎ / 𝑓](𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓)))
16		bnj252 31973	. . . . . 6 ⊢ ((𝑁 ∈ 𝐷 ∧ ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓) ↔ (𝑁 ∈ 𝐷 ∧ (ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓)))
17	14, 15, 16	3bitr4ri 306	. . . . 5 ⊢ ((𝑁 ∈ 𝐷 ∧ ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓) ↔ [ℎ / 𝑓](𝑁 ∈ 𝐷 ∧ (𝑓 Fn 𝑁 ∧ 𝜑 ∧ 𝜓)))
18	5, 7, 17	3bitr4i 305	. . . 4 ⊢ ([ℎ / 𝑓]𝜒 ↔ (𝑁 ∈ 𝐷 ∧ ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓))
19		fneq1 6444	. . . . . . 7 ⊢ (ℎ = 𝐺 → (ℎ Fn 𝑁 ↔ 𝐺 Fn 𝑁))
20		sbceq1a 3783	. . . . . . . 8 ⊢ (ℎ = 𝐺 → ([ℎ / 𝑓]𝜑 ↔ [𝐺 / ℎ][ℎ / 𝑓]𝜑))
21		bnj976.2	. . . . . . . . 9 ⊢ (𝜑′ ↔ [𝐺 / 𝑓]𝜑)
22		sbccow 3795	. . . . . . . . 9 ⊢ ([𝐺 / ℎ][ℎ / 𝑓]𝜑 ↔ [𝐺 / 𝑓]𝜑)
23	21, 22	bitr4i 280	. . . . . . . 8 ⊢ (𝜑′ ↔ [𝐺 / ℎ][ℎ / 𝑓]𝜑)
24	20, 23	syl6bbr 291	. . . . . . 7 ⊢ (ℎ = 𝐺 → ([ℎ / 𝑓]𝜑 ↔ 𝜑′))
25		sbceq1a 3783	. . . . . . . 8 ⊢ (ℎ = 𝐺 → ([ℎ / 𝑓]𝜓 ↔ [𝐺 / ℎ][ℎ / 𝑓]𝜓))
26		bnj976.3	. . . . . . . . 9 ⊢ (𝜓′ ↔ [𝐺 / 𝑓]𝜓)
27		sbccow 3795	. . . . . . . . 9 ⊢ ([𝐺 / ℎ][ℎ / 𝑓]𝜓 ↔ [𝐺 / 𝑓]𝜓)
28	26, 27	bitr4i 280	. . . . . . . 8 ⊢ (𝜓′ ↔ [𝐺 / ℎ][ℎ / 𝑓]𝜓)
29	25, 28	syl6bbr 291	. . . . . . 7 ⊢ (ℎ = 𝐺 → ([ℎ / 𝑓]𝜓 ↔ 𝜓′))
30	19, 24, 29	3anbi123d 1432	. . . . . 6 ⊢ (ℎ = 𝐺 → ((ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓) ↔ (𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′)))
31	30	anbi2d 630	. . . . 5 ⊢ (ℎ = 𝐺 → ((𝑁 ∈ 𝐷 ∧ (ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓)) ↔ (𝑁 ∈ 𝐷 ∧ (𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′))))
32		bnj252 31973	. . . . 5 ⊢ ((𝑁 ∈ 𝐷 ∧ 𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑁 ∈ 𝐷 ∧ (𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′)))
33	31, 16, 32	3bitr4g 316	. . . 4 ⊢ (ℎ = 𝐺 → ((𝑁 ∈ 𝐷 ∧ ℎ Fn 𝑁 ∧ [ℎ / 𝑓]𝜑 ∧ [ℎ / 𝑓]𝜓) ↔ (𝑁 ∈ 𝐷 ∧ 𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′)))
34	18, 33	syl5bb 285	. . 3 ⊢ (ℎ = 𝐺 → ([ℎ / 𝑓]𝜒 ↔ (𝑁 ∈ 𝐷 ∧ 𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′)))
35	3, 34	sbcie 3812	. 2 ⊢ ([𝐺 / ℎ][ℎ / 𝑓]𝜒 ↔ (𝑁 ∈ 𝐷 ∧ 𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′))
36	1, 2, 35	3bitr2i 301	1 ⊢ (𝜒′ ↔ (𝑁 ∈ 𝐷 ∧ 𝐺 Fn 𝑁 ∧ 𝜑′ ∧ 𝜓′))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3494  [wsbc 3772   Fn wfn 6350   ∧ w-bnj17 31956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-fun 6357  df-fn 6358  df-bnj17 31957

This theorem is referenced by:  bnj910  32220  bnj999  32230  bnj907  32239