Theorem bnj581 35404

Description: Technical lemma for bnj580 35409. 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 3798	. 2 ⊢ ([𝑔 / 𝑓]𝜒 ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		sbc3an 3806	. . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓))
5		bnj62 35217	. . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛)
6	5	bicomi 227	. . . 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 1173	. . 3 ⊢ ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓))
10	4, 9	bitr4i 281	. 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
11	1, 3, 10	3bitri 300	1 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1103  [wsbc 3742   Fn wfn 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-fun 6539  df-fn 6540

This theorem is used by:  bnj580  35409  bnj849  35421