Theorem bnj156 35126

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

Hypotheses

Ref	Expression
bnj156.1	⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
bnj156.2	⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)
bnj156.3	⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
bnj156.4	⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)

Assertion

Ref	Expression
bnj156	⊢ (𝜁1 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))

Proof of Theorem bnj156

Step	Hyp	Ref	Expression
1		bnj156.2	. 2 ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0)
2		bnj156.1	. . . 4 ⊢ (𝜁0 ↔ (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
3	2	sbcbii 3799	. . 3 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ [𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
4		sbc3an 3807	. . . 4 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1o ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′))
5		bnj62 35118	. . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 1o ↔ 𝑔 Fn 1o)
6		bnj156.3	. . . . . 6 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
7	6	bicomi 227	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ 𝜑1)
8		bnj156.4	. . . . . 6 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
9	8	bicomi 227	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ 𝜓1)
10	5, 7, 9	3anbi123i 1172	. . . 4 ⊢ (([𝑔 / 𝑓]𝑓 Fn 1o ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
11	4, 10	bitri 278	. . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
12	3, 11	bitri 278	. 2 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
13	1, 12	bitri 278	1 ⊢ (𝜁1 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))

This proof depends on syntax axioms:   ↔ wb 209   ∧ w3a 1102  [wsbc 3743   Fn wfn 6531  1_oc1o 8444

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-fun 6538  df-fn 6539

This theorem is used by:  bnj153  35277