Theorem bnj156 31314

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 3689	. . 3 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ [𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′))
4		sbc3an 3691	. . . 4 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1𝑜 ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′))
5		bnj62 31306	. . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 1𝑜 ↔ 𝑔 Fn 1𝑜)
6		bnj156.3	. . . . . 6 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
7	6	bicomi 216	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ 𝜑1)
8		bnj156.4	. . . . . 6 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
9	8	bicomi 216	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ 𝜓1)
10	5, 7, 9	3anbi123i 1195	. . . 4 ⊢ (([𝑔 / 𝑓]𝑓 Fn 1𝑜 ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))
11	4, 10	bitri 267	. . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))
12	3, 11	bitri 267	. 2 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))
13	1, 12	bitri 267	1 ⊢ (𝜁1 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1))

Syntax hints:   ↔ wb 198   ∧ w3a 1108  [wsbc 3633   Fn wfn 6096  1_𝑜c1o 7792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-fun 6103  df-fn 6104

This theorem is referenced by:  bnj153  31467