Theorem bnj581 30232

Description: Technical lemma for bnj580 30237. 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). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-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 3458	. 2 ⊢ ([𝑔 / 𝑓]𝜒 ↔ [𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		sbc3an 3461	. . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓))
5		bnj62 30040	. . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛)
6	5	bicomi 213	. . . 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 1244	. . 3 ⊢ ((𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 𝑛 ∧ [𝑔 / 𝑓]𝜑 ∧ [𝑔 / 𝑓]𝜓))
10	4, 9	bitr4i 266	. 2 ⊢ ([𝑔 / 𝑓](𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))
11	1, 3, 10	3bitri 285	1 ⊢ (𝜒′ ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))

Syntax hints:   ↔ wb 195   ∧ w3a 1031  [wsbc 3402   Fn wfn 5799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807

This theorem is referenced by:  bnj580  30237  bnj849  30249