Theorem bnj118 35433

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj118.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj118.2	⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)

Assertion

Ref	Expression
bnj118	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛

Allowed substitution hints:   𝜑(𝑥, 𝑓, 𝑛)   𝐴(𝑥, 𝑓)   𝑅(𝑥, 𝑓)   𝜑′(𝑥, 𝑓, 𝑛)

Proof of Theorem bnj118

Step	Hyp	Ref	Expression
1		bnj118.2	. 2 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
2		bnj118.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3		bnj105 35289	. . 3 ⊢ 1o ∈ V
4	2, 3	bnj91 35425	. 2 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5	1, 4	bitri 278	1 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  [wsbc 3738  ∅c0 4278  ‘cfv 6527  1_oc1o 8447   predc-bnj14 35253

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 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-pw 4558  df-sn 4584  df-suc 6357  df-1o 8454

This theorem is used by:  bnj151  35441  bnj153  35444