Theorem bnj526 35384

Description: Technical lemma for bnj852 35417. 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
bnj526.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj526.2	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
bnj526.3	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
bnj526	⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

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

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

Proof of Theorem bnj526

Step	Hyp	Ref	Expression
1		bnj526.2	. 2 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
2		bnj526.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3	2	sbcbii 3798	. 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ [𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4		bnj526.3	. . 3 ⊢ 𝐺 ∈ V
5		fveq1 6881	. . . 4 ⊢ (𝑓 = 𝐺 → (𝑓‘∅) = (𝐺‘∅))
6	5	eqeq1d 2764	. . 3 ⊢ (𝑓 = 𝐺 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
7	4, 6	sbcie 3783	. 2 ⊢ ([𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
8	1, 3, 7	3bitri 300	1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  Vcvv 3453  [wsbc 3742  ∅c0 4282  ‘cfv 6537   predc-bnj14 35185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743  df-ss 3919  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545

This theorem is used by:  bnj607  35412