Theorem bnj156 34704

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 3865	. . 3 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ [𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))
4		sbc3an 3874	. . . 4 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1o ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′))
5		bnj62 34696	. . . . 5 ⊢ ([𝑔 / 𝑓]𝑓 Fn 1o ↔ 𝑔 Fn 1o)
6		bnj156.3	. . . . . 6 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
7	6	bicomi 224	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ 𝜑1)
8		bnj156.4	. . . . . 6 ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′)
9	8	bicomi 224	. . . . 5 ⊢ ([𝑔 / 𝑓]𝜓′ ↔ 𝜓1)
10	5, 7, 9	3anbi123i 1155	. . . 4 ⊢ (([𝑔 / 𝑓]𝑓 Fn 1o ∧ [𝑔 / 𝑓]𝜑′ ∧ [𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
11	4, 10	bitri 275	. . 3 ⊢ ([𝑔 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
12	3, 11	bitri 275	. 2 ⊢ ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))
13	1, 12	bitri 275	1 ⊢ (𝜁1 ↔ (𝑔 Fn 1o ∧ 𝜑1 ∧ 𝜓1))

Syntax hints:   ↔ wb 206   ∧ w3a 1087  [wsbc 3804   Fn wfn 6568  1_oc1o 8515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-fun 6575  df-fn 6576

This theorem is referenced by:  bnj153  34856