Theorem bnj581 32884

Description: Technical lemma for bnj580 32889. 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.) Remove unnecessary distinct variable conditions. (Revised by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj581.3	⊢ (𝜒 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj581.4	⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑)
bnj581.5	⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓)
bnj581.6	⊢ (𝜒′ ↔ [𝑔 / 𝑓]𝜒)

Assertion

Ref	Expression
bnj581	⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))

Distinct variable group:   𝑓,𝑛

Allowed substitution hints:   𝜑(𝑓,𝑔,𝑛)   𝜓(𝑓,𝑔,𝑛)   𝜒(𝑓,𝑔,𝑛)   𝜑′(𝑓,𝑔,𝑛)   𝜓′(𝑓,𝑔,𝑛)   𝜒′(𝑓,𝑔,𝑛)

Proof of Theorem bnj581

Step	Hyp	Ref	Expression
1		bnj581.6	. 2 ⊢ (𝜒′ ↔ [𝑔 / 𝑓]𝜒)
2		bnj581.3	. . 3 ⊢ (𝜒 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
3	2	sbcbii 3781	. 2 ⊢ ([𝑔 / 𝑓]𝜒 ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		sbc3an 3791	. . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓))
5		bnj62 32695	. . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛)
6	5	bicomi 223	. . . 4 ⊢ (𝑔 Fn 𝑛 ↔ [𝑔 / 𝑓]𝑓 Fn 𝑛)
7		bnj581.4	. . . 4 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑)
8		bnj581.5	. . . 4 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓)
9	6, 7, 8	3anbi123i 1154	. . 3 ⊢ ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓))
10	4, 9	bitr4i 277	. 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
11	1, 3, 10	3bitri 297	1 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))

Syntax hints:   ↔ wb 205   ∧ w3a 1086  [wsbc 3720   Fn wfn 6427

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 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-fun 6434  df-fn 6435

This theorem is referenced by:  bnj580  32889  bnj849  32901