Theorem bnj526 34924

Description: Technical lemma for bnj852 34957. 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 3827	. 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ [𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4		bnj526.3	. . 3 ⊢ 𝐺 ∈ V
5		fveq1 6880	. . . 4 ⊢ (𝑓 = 𝐺 → (𝑓‘∅) = (𝐺‘∅))
6	5	eqeq1d 2738	. . 3 ⊢ (𝑓 = 𝐺 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
7	4, 6	sbcie 3812	. 2 ⊢ ([𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
8	1, 3, 7	3bitri 297	1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3464  [wsbc 3770  ∅c0 4313  ‘cfv 6536   predc-bnj14 34724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-sbc 3771  df-ss 3948  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544

This theorem is referenced by:  bnj607  34952