Theorem bnj609 32299

Description: Technical lemma for bnj852 32303. 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). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj609.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj609.2	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
bnj609.3	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
bnj609	⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑓,𝑋

Allowed substitution hints:   𝜑(𝑓)   𝐺(𝑓)   𝜑″(𝑓)

Proof of Theorem bnj609

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj609.2	. 2 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
2		bnj609.3	. . 3 ⊢ 𝐺 ∈ V
3		dfsbcq 3722	. . 3 ⊢ (𝑒 = 𝐺 → ([𝑒 / 𝑓]𝜑 ↔ [𝐺 / 𝑓]𝜑))
4		fveq1 6644	. . . 4 ⊢ (𝑒 = 𝐺 → (𝑒‘∅) = (𝐺‘∅))
5	4	eqeq1d 2800	. . 3 ⊢ (𝑒 = 𝐺 → ((𝑒‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
6		bnj609.1	. . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7	6	sbcbii 3776	. . . 4 ⊢ ([𝑒 / 𝑓]𝜑 ↔ [𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8		vex 3444	. . . . 5 ⊢ 𝑒 ∈ V
9		fveq1 6644	. . . . . 6 ⊢ (𝑓 = 𝑒 → (𝑓‘∅) = (𝑒‘∅))
10	9	eqeq1d 2800	. . . . 5 ⊢ (𝑓 = 𝑒 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)))
11	8, 10	sbcie 3760	. . . 4 ⊢ ([𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
12	7, 11	bitri 278	. . 3 ⊢ ([𝑒 / 𝑓]𝜑 ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
13	2, 3, 5, 12	vtoclb 3512	. 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
14	1, 13	bitri 278	1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3441  [wsbc 3720  ∅c0 4243  ‘cfv 6324   predc-bnj14 32068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721  df-in 3888  df-ss 3898  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332

This theorem is referenced by:  bnj600  32301  bnj908  32313  bnj934  32317