Theorem bnj156 31611

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 3763	. . 3 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ [𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
4		sbc3an 3772	. . . 4 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1o ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′))
5		bnj62 31603	. . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 1o ↔ 𝑔 Fn 1o)
6		bnj156.3	. . . . . 6 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
7	6	bicomi 225	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ 𝜑1)
8		bnj156.4	. . . . . 6 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
9	8	bicomi 225	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ 𝜓1)
10	5, 7, 9	3anbi123i 1148	. . . 4 ⊢ (([𝑔 / 𝑓]𝑓 Fn 1o ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
11	4, 10	bitri 276	. . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
12	3, 11	bitri 276	. 2 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
13	1, 12	bitri 276	1 ⊢ (𝜁1 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))

Syntax hints:   ↔ wb 207   ∧ w3a 1080  [wsbc 3711   Fn wfn 6227  1_oc1o 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-fun 6234  df-fn 6235

This theorem is referenced by:  bnj153  31764