Theorem bnj526 30212

Description: Technical lemma for bnj852 30245. 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 3458	. 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ [𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
4		bnj526.3	. . 3 ⊢ 𝐺 ∈ V
5		fveq1 6102	. . . 4 ⊢ (𝑓 = 𝐺 → (𝑓‘∅) = (𝐺‘∅))
6	5	eqeq1d 2612	. . 3 ⊢ (𝑓 = 𝐺 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
7	4, 6	sbcie 3437	. 2 ⊢ ([𝐺 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
8	1, 3, 7	3bitri 285	1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ∅c0 3874  ‘cfv 5804   predc-bnj14 30007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by:  bnj607  30240