Theorem bnj609 34679

Description: Technical lemma for bnj852 34683. 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 3775	. . 3 ⊢ (𝑒 = 𝐺 → ([𝑒 / 𝑓]𝜑 ↔ [𝐺 / 𝑓]𝜑))
4		fveq1 6895	. . . 4 ⊢ (𝑒 = 𝐺 → (𝑒‘∅) = (𝐺‘∅))
5	4	eqeq1d 2727	. . 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 3834	. . . 4 ⊢ ([𝑒 / 𝑓]𝜑 ↔ [𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8		vex 3465	. . . . 5 ⊢ 𝑒 ∈ V
9		fveq1 6895	. . . . . 6 ⊢ (𝑓 = 𝑒 → (𝑓‘∅) = (𝑒‘∅))
10	9	eqeq1d 2727	. . . . 5 ⊢ (𝑓 = 𝑒 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)))
11	8, 10	sbcie 3817	. . . 4 ⊢ ([𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
12	7, 11	bitri 274	. . 3 ⊢ ([𝑒 / 𝑓]𝜑 ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
13	2, 3, 5, 12	vtoclb 3547	. 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
14	1, 13	bitri 274	1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3461  [wsbc 3773  ∅c0 4322  ‘cfv 6549   predc-bnj14 34450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-sbc 3774  df-ss 3961  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557

This theorem is referenced by:  bnj600  34681  bnj908  34693  bnj934  34697