Theorem bnj1040 32523

Description: Technical lemma for bnj69 32561. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1040.1	⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑)
bnj1040.2	⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)
bnj1040.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1040.4	⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒)

Assertion

Ref	Expression
bnj1040	⊢ (𝜒′ ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))

Distinct variable groups:   𝐷,𝑖   𝑓,𝑖   𝑖,𝑛

Allowed substitution hints:   𝜑(𝑓,𝑖,𝑗,𝑛)   𝜓(𝑓,𝑖,𝑗,𝑛)   𝜒(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑗,𝑛)   𝜑′(𝑓,𝑖,𝑗,𝑛)   𝜓′(𝑓,𝑖,𝑗,𝑛)   𝜒′(𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1040

Step	Hyp	Ref	Expression
1		bnj1040.4	. 2 ⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒)
2		bnj1040.3	. . 3 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
3	2	sbcbii 3738	. 2 ⊢ ([𝑗 / 𝑖]𝜒 ↔ [𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		df-bnj17 32236	. . 3 ⊢ (([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑 ∧ [𝑗 / 𝑖]𝜓) ↔ (([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓))
5		vex 3402	. . . . . 6 ⊢ 𝑗 ∈ V
6	5	bnj525 32288	. . . . 5 ⊢ ([𝑗 / 𝑖]𝑛 ∈ 𝐷 ↔ 𝑛 ∈ 𝐷)
7	6	bicomi 227	. . . 4 ⊢ (𝑛 ∈ 𝐷 ↔ [𝑗 / 𝑖]𝑛 ∈ 𝐷)
8	5	bnj525 32288	. . . . 5 ⊢ ([𝑗 / 𝑖]𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛)
9	8	bicomi 227	. . . 4 ⊢ (𝑓 Fn 𝑛 ↔ [𝑗 / 𝑖]𝑓 Fn 𝑛)
10		bnj1040.1	. . . 4 ⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑)
11		bnj1040.2	. . . 4 ⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓)
12	7, 9, 10, 11	bnj887 32315	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑 ∧ [𝑗 / 𝑖]𝜓))
13		df-bnj17 32236	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝜓))
14	13	sbcbii 3738	. . . 4 ⊢ ([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ [𝑗 / 𝑖]((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝜓))
15		sbcan 3730	. . . 4 ⊢ ([𝑗 / 𝑖]((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝜓) ↔ ([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ [𝑗 / 𝑖]𝜓))
16		sbc3an 3747	. . . . 5 ⊢ ([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑))
17	16	anbi1i 627	. . . 4 ⊢ (([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ [𝑗 / 𝑖]𝜓) ↔ (([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓))
18	14, 15, 17	3bitri 300	. . 3 ⊢ ([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓))
19	4, 12, 18	3bitr4ri 307	. 2 ⊢ ([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
20	1, 3, 19	3bitri 300	1 ⊢ (𝜒′ ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   ∈ wcel 2114  [wsbc 3680   Fn wfn 6334   ∧ w-bnj17 32235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-sbc 3681  df-bnj17 32236

This theorem is referenced by:  bnj1128  32541